J/W^rm''\tz^ jON NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1224 lAME'S WAVE FUNCTIONS OF THE ELULPSOID OF REVOLUTION By J. Meixner Translation of ZWB Forschungsbericht Nr. 1952, June 1944 Washington April 1949 ^-fj ^i^O.. -^-^o^\i IJAGA TM No. 1224 TABLE OF CONTENTS Page 1. BffBODUCTIOK . ■ , 1 2 . THE BASIC EQ^TIOW 3 2.1 Eotationally. Symmetrical Elliptic Coordiriabis ... 3 2.2 Separation of tlie I'T'ave Eqiuation 4 2.3 Eeduction to. a Differential Eq.uetion 5 ^..k- Transformations of the Basic Eciuation 5 2.5 Connection with Mathieu's Eunctione 6 3. SPEESICAL iaSCYLlIffiPJC/l, FUHCTIOHS 7 3.1 A Few Fonaulas for Spherical Fimctions 7 • 3-2 A Few Estimates for- SphericixL Functions 8 3.3 A' Few Formialas for Cylindr-ical F^jnctlons 10 3-^ An Estimate for Cylindrical Functions 11 !+. THE X-FimTCTIONE OF THE FIRST PMs SSCOi\!D KIHD • I3 'j-.l Definition of the Z"Functiona of the First and Second Kind I3 h.2 General Qualities of the Coefficients a^, . . . . . 15 4.3 Convergence of the Series Developments of the Z-Functions of the First and Second Kind .... I6 h-k Further Solutions of the Basic Equation and Their Relation to the X-Functions of the First and Second Kind ' IT 4.5 General Kelations between the X-Functions 20 5. TEE Z-5TJIICTI0KS OF TBE.. FIRST TO FOURTH KIIfD • 21 5-1 Defixiition of the Z -Functions of the First to I'o'oi-th Kind 21 5-2 Convergence of the Series Developraexits of the Z~Functions ' of the First and Second Kind .... 22 5-3 General Eelations hetwecn ohe Z-Fimctions 23 5.^ Asymptotic Developments of the Z-Functions 2k 5-5 Fi;irtLer Solutions of the Basic Equation and Their Eelation to this Z-Fimctions of the First and Second Kind ............... 27 5-6 Laurent •'Developments for X- and Z-F-unctions .... 28 5-7 Connection hetween the X- and Z -Functions 3I 5-8 Wronski 'b' Determinant . .' 33 t>-.9 Other Series Developments of the Solutions of the Basic Equation 3i|- NAGA TM No. 122L Page 6. GV\LCUL/^riON OF THE COEFFICIENI'S OF THE SEHUS DE^.TCLOPMEifTS DJ TEESvfS OF SPBERICi\L .^m CyLIKLEICAL FIMCTIOKS ..... 37 6.1 Continued Fi-action D©Yc;lopnents 37 6.2 M&ohod for Numerical Calculation of the Separation Paraineter and the Development CoeffiGieiif;8 .... 38 6.3 Power Series for Separation Parameter and Devel0]jsnent Coefficients 39 6. k Poorer Series Developments kk 7. EIGEIJFTO'jCTIOWS of the BASIC EQUATION 47 7-1 Limitation to v ,|i Being Integers^ v -ll^ii- • • . 47 7-2 Brealring Off of the Series hQ 7-3 A ^''ev Special Function Values 50 7-4 Connection Detween the X- and Z-Functions 51 7' 5 Kormalization mA Properties of Or'thogonality of the X -Functions of the First Kind 53 7' 6 Generalization of F. F- Ife^anann's Integral Relation . 5^ 7-T Zeros of the Eis;enftaictions 55 7-8 In^gral Equations for the Eigenfujictions 56 ,£- ASyiviPTOTICS OF TEE EIGEW/iLUES AIID EIGEl^'FUI'ICTIONS 57 8.1 Asyniptotic Behavior of the Eigenvalues and LigenfujQctions for Large V 57 8.2 A3;iTaptotic Behavior of the Eigenvalues for Large Peal 7 • 59 8.3 Asymptotic Behavior of the Eigenfixrictions for Large Peal 7 61 8.14- Asyiiptotic Behavior of the Eigenvalues for T,arge Purelj?- Imaginary 7 63 8.5 Asymptotic Behavior of the Eigenf unctions for Large Purely Imaginary 7 65 9- EIGEWFIMCTIOIMS OF THE WAVE EQUATION IN EOTATIONi'XLY SyM4!?rBICAL ELLIPTIC COOPDIN/iTES 67 . 9-1 Larae's Wave Functions of the Prolate Elllpacid of Revolution 67 . .9-2, Lame's Wave Functions of the Ohlate Ellipsoid of Pevolution ■ - 69 9-3 Normalization of Lame's Wave Functions for Outside Space Prohlems 70 9-k Development of Lamp's Wave Fvjictions in Tei'ins of Spherical and Cylindrical i;taact ions 72 mCA TM Ko. 1224. Pag© 10. THE MEJi'HOlJ OF GREEK'S rUwO-aMON FOP. TBE SOLOTIOE OF BOUITO/1BY YALTJE KtOBLEiC, PAHTICUT./*LI OF RADIATION PEOSLE'IS ... 74 10.1 Green's Function of the ¥ave Equation in Radiation ProDleins 7': 10.2 Development of the Spherical Wave and of the- ^lane Wave in Tei'ae of Lacie's Ware .?\vrj.Cwi-ir.,-i .... 76 10.3 Diffraction of a Scalar Sphe-iJc'-j, Vkwe or Plane Wave on the Ellipsoid of Ee'volution 78 11, TABIES 80 11-1 Coiniaents to the Ta'blee 80 11.2 Eigenvalues X (7) and Devolopmont Coeffi- m . . m , . cients a (7)^ h .(7/> Eepi-eBonted "by Brokon- Off Povor Sorios in 7 85 11.3 Numerical Magnitude of the Eigenvaluos and the Devo I op-Blunt Coefficients for Different n.,7 and m = 95 11. 4- Course, of tho Curv g X - X^iy) for Lov Values of tho Indox n 102 Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A.'Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/lameswavefunctioOOunit NATIONAL ADVISOEY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1224 urn's WAVE FUNCTIONS OF TBE ELLXPSOID OF EE^raLUTION* By J. Meixner 1. INTROroCTION Lame's wave ftaictions reEuLt by separation of the wave equation in elliptic coordinates and "by integration of the ordinary differen- tial equations thus originating. They are a general-ization of Lame's potential functions which originate in the same nanner from the potential equation. Lame's wave functions are applied for boundary value problems of the wave equation for regions of space bounded by surfaces of a system of confocal ellipsoids and hyperboloids . For general elliptic coordinates Lame's wave functions ^ave not been fully calculated so far. Except for a few general properties, not much is Imown about them. More consideration was given to Leme's wave ftmctions for the case of rotationally Dymraetrical elliptic coordinates (called for short, Lame's wave f-onctions of the ellipsoid of revolution) . However, even for these functions few results are in existence compared with those for the better I'Oao'VJn special functions of mathematical physics, such as cylindi'ical and spherical functions. The first mors detailed investigation of Lame's wave fianctions of the ellipsoid of revolution was made by Niven (reference l) who with their aid treated a heat- conduction problem in the ellipsoid of revolution. Howeiver, the numerical values of the coefficients of his series developments in terms of spherical and cylindrical functions as they are given for the lowest indices contain several errors which were taken over into the report by Stiaitt (reference 2) , A more extensive investigation with a greater nimbcr of applications was made by Maclaurln (reference 3) • . Mogllch (reference h) , whoee mathematical investigation of Lame''s wave equation is based on certain linear homogeneous integral equations, obtained results of a Die Lameschen Wellenf unktionen des Drehellipsolds . Zentrale five wissenschaftlicheB Berichtswesen der Luf tf ahrtfcrschung des Generalluftzeugmeisters (ZWB) Berlin- Adlershof, Forschvingsbericht Nr. 1952, June 19'+'+ • MCA TM No. 1224 more genere-l character. Strutt (reference 2) gives a survey of the state of the theory of Lame's -.ra,ve functions in 1932; he also demon- strates on a lai'ge niMber of examples from acoustics, electrodynamics, optics, wave mechanics, and theory of wave filters, the manifold possibilities of application for these f -unctions. ' Of the treatises published in the meantime, an investigation hy Eanson (reference 5), which contains several new details, should he mentioned, as well as a treatise 'by Morse (reference 6) on addition theorems, that is, on the development of the plane wave and tlie spherical wave in terms of Lame's wave functions, furthermore, a number of ti-eatises on the wave -mechanical treatment of the ion of the hydrogen molec^ole (reference 7) • Kotani (reference 8) deals with integral equations for Lame's wave functions. In particular, a treatise by Chu and Stratton (reference 9) should be pointed out -t-Siich settles exhaustively the problem (treated so f.ar only incompletely) of the continuation of the solutions of equation (2. kg) for large and small argument and shows in detail how the entire theory of Mathieu's functions resijlts as a special and boundary case x'rom the general theory of Lame's wave functions. Finally, a treatise by Botxwkamp (reference 10) on the theoretical and numerical treatment of diffraction on a circular apertiire is to be mentioned which, for the first time, contains more detailed numerical material concerning Lame''s wave functions of the ellipsoid of revolution. . , The main task of the present report on Lame's wa\'-e fimctions of the ellipsoid of revolution will be to compile their most important properties in such a manner that these functions take on a form which facilitates their application. In this connection an investigation of the solutions of the ordinary homogeneous linear differential equations of the second order, which originate with separation of the wave equation in rotationally symmetrical elliptic coordinates, is of importance; further, it has to be determined what is to be understood in these solutions by fujnctions of the first and seconc'. kind, their normalization as well as the description of the behavior of these solutions in different domains of the independent variables, in particular', their as^rmptotic behavior. Here belongs also the indication of a method of numerical calculation of these functions and the presentation of numerical tables. For the pu3rpose of clarity it was necessary to generalize and supplement the existing material in some respects and to simplify some of the calculations and proofs. Therewith the theory of Lane's wave functions of the ellipsoid of revolution as a vrhole would seem to have reached a development equivalent to the theoi-y of Mathieu's MCA TM No. 122i<- -p a m 5 03 g •H •P Q CO m •H P H 13 & e m -P i n3 H^ (D O O O (D 0) Pi >s OQ a rd CO O: ti cd o CO o Pi CVl CO H CJ •1/1 o CO OJ CO 1 H ID O O CVJ OJ H OJ V" V" o V" V" "NP ;^ V H CO o itn -5 P> ^ ,' S II V" U Pi o ca o « O II rH I CM ^•-^ CJ cr I H O II 03 O o H I OJ OJ I H O II OJ V" e- V" o H VII p- VII H 1 H H in CM CiD P> P 02 Q O P> fl O •H +3 5J ?S c © p! o •H P CO pi (D » o g •H ■P CO f^ Cj P) K -P PI •H ,jf^ = (2 .5a) A ^1-1^)?!^ d^ 1 - 1^ ,2*2 \ k'^c'^l- + 7JfT = y^ (2.1tg) g dn dn 1 - n'^ ^2 2„2 \jfo = (2.5s) d9' 2 - -. 2f ,2 " ^ 3 (2.6) are valid for f-j_, fp, ^md fo' A, and y.- are the separation parameters. Hirst ^ they are assimed to he any complex n-iBabei'S . Hiey can on3.y he determined for a given boundary value prohlen. In particular, |i need not MCA TM Wo. 122ll. be an integer; this can be recognized^ for instance^ in the treat- ment of an inside space problem in a sector = cp = cp of an ellipsoid of revolution. 2,3 Eeduction to a Differential 'illqu.ation The differential equation (2.J-i-g) is designated as the basic equation. (2.5g) is identical with it;_^(2 .ii-a) is transformed' into it when I is replaced by ±il and k^c^ by -1:^0^-. Therewith the investigation of the differential equations (2 ,ha) , (2 .5a) , and (2.5g) is reduced to that of the differential equation (2 .4g) . The basic domain^ however, is not the same for all cases; it extends from -1 to 1 in the cases (2.5a) and (2.5e), from 1 to <» in the case (2 .4g) , whereas the basic domain of the differential equation (2.1j.a) in the transformation to (2.ii-g) mil be changed to the domain from to i<» (or .'a-lse -i»). It proves , therefore, to be necessary to investigate the differential equation (2.)4-g) in the entire complex \ -"olajie . 2.^ Transformations of the Basic Equation The basic equation represents a sioecial case of the linear homogeneous differential equation of the second order with fovx extra essential singularities, two of which are made to join to one essential singularity. The latter is at infinity, the two remaining extra essential singularities are at 1 and -1. The present investigation of the basic equation will start with connecting its solrtions with the solutions of limiting cases of the basic equation. For l0. From {'2- .kg) there originates with fi = (i^ - /T'" r^ v(o (2.9) the differential equation {}2 .^2^^+ 2^A ., J..t 2\dY ^..,.)^.,^h,^,^g ...^a.tiltL^^ J V = (2.10) In the transition from (2 .1+g) to (2 Aa) ^ is transf ormeci into itself and 7^ need only be replaced by -7 . For large distances^ 90000 " ■ that is, r'" = x"" + y'" + z » c ?'2 _ ,,2,;2 I;ij_£)^ , of =f Vr"''/ ~.^ r (2.11a) ^2 = kV /.U + ^^ -^/)^" + of'^- Vr^> (2.11g) are valid. ^toother irnportant limiting case of the basic equ.ation occnrs if, of the two singularities of the basic equation located at finite distance, one or both move to infinity. Then the differential equation of Laguerre's and Hermite's orthogonal functions, respec- tively, is formed. This limiting case will yield the asymptotics of the eigenvalues and eigenf uJictions for large absolute value of 7 - 2.5 Connection with Mathieu's Functions Mathieu's functions are, in connection with Lame's wave functions, obtained in two ways. They appear, as is well icnovai, in the separa- tion of the wave equation in the coordinates of the elliptic cylinder and must, therefore, also appear in the limiting case of Lame's wave functions for the ellipsoid with three axes when one axis beconies MCA TM Wo. 12214- infinitely long. However, Mathieti's differential eqiiation is also obtained, except for an elementary transformation, if [i in (2.)+g) is set equal to tl/2 . This also indicates that it is tiseful to consider the hasic equation not on.ly for |i that are integers, hut rather for arbitrary coefficients V and u. The theory of Mathieu's fvmctions is, therefore, a special case of the theory of Lame's I'ave fimctions of the ellipsoid of revolution. Although the present report does not yield new results of ]\Iathieu's functions, it demonstrates how they fit into a more general picture . 3 . SPHSEICAL AND CYLIMDEICAL ITOTGTIONS 3*1 A Few Formulas for Spherical Functions The most impoi-tant formulas and theorems for sp)herical and cylindric functions needed below are compiled and a few estimates for these functions are given, irhich will be necessary for con- siderations on uxiiform convei^gence of certain series in terms of such functions. Magnus and Oberhettinger (reference H) is again referred to concerning the notation and additional formulas. Tlie general spherical functions differential equation Pt}(^) and 9H'(I) both satisfy the A. dl (1 n IV (V + 1) - ^2 pH(^) = (3.1) and both satisfy the recursion f oimiula (2v + i)ip{;(o = (V - ^ -•- DPy+^a) + (v + n)p{^.^(i) (3.2) from which by three times repeated application .2pH/.') _ .(L_-_iL±_2l(V_J_li_t_D.pH us .. g yg + 2V - 2^2 - 1 ^,. ^ iv^^^ ~ (2V + 1)(2V + 3) £v+2^'-> ' (2v - l) (2v + 3) "-V ''^ ^ +.(V-^ k) (V ■!- H- - l)p H (.) (2V - 1)(2V -;- 1) -V-2^^^ (3-3) MCA TM No. 1224 m n . U/l 1 p< o OJ • • 1 A a g iH P ,*— s o •H 1 t> . iin t:! fn •P •H- OJ S^ ^ "i? y— > •ti d 1 ra ^— s H H ;3 Oi •ri H + ii - H H + fH m in A P- •H Pi :i Ph o + . OJ C "H +3 H >_ /^7 ^ s CS p! 4Ln + oolcj b fe ^..^ d. + rC5 O O © +3 1 o f^ + P Pi a -P fl .\ O O CO O •H 1 Ch + ^M fi P © H CQ t-, ^ [^ ^ + $Pa -'' kj +3 o t5D C_ o f-i ►l 1h cd n:) + © II a o P! 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I In" d. ?• P-ii 10 WACA TM IJo. 122i| It is valid for -2n < afg z'^ < 2jt with the provision that the path of integration for jt < arg z < 2-a and -2jt < arg z < -rt, respec- tively, leads past the left of the point u = and is also to "be retiimed there. The estimate has to "be made as alcove. The maxiiavm of |u|^ is, if the path of integration is suitably selected, smaller than (izl^ + 6)-"^ for positive r's and smaller than b^ for negative r's, where S is a number of the conditions indicated above, so that . P^ (I) < l\z\ + -^) ^^(i) for r = 0, 1, 2, . . . v+r y ' \z\J V (3.8) = [zj'^t "^^0 ■ f°^ ^' = -1^ -2. "3, . • . respectively, where i^{i) and i|J(^) oi-e positive, rest].-icted functions independent of r in each closed domain excluding the points I = ±l,oD . 3-3 A Few Formulas for Cylindrical Functions For the following it is more convenient to introduce not the cylinOjr-ical functions themselves but rather the functions %^^^ -U^^l/^^^^' ^v^^ ) = yj \n/2 (^ ) (3 .9) They both satisfy the differential equation V(V 4- 1) £!v,. 2 5^ a.r" ^ d^ tv = (3.10) NACA TM No. 122i^ 11 and the recursion fomujlas 2v + 1 ^v V ,,, , £v + 1 ,y V -i- 1 t /. ,,N i d^ ~ 2v - 1 '^-2 (ov-i)(2v+3) V 2V + 3 v+2 ^^*^^^ -V + 1 )lr L_ ^^ + 2.(.2V ,-»:. 1.) ■— . ^\, + .__!_ x^ (-, .12) ^2 V - 2V - 1 V-2 (2v - 1)(2V + 3) V 2V + 3 V+2 "^^'^ ' Besides, the simpler recursion fomula f\ii) = Vi^U -^ ^,-i(0 (3.13) is to "be noted for the cylindrical fxmctions from which (3-12) is ohtained hy repeated app].lcatlon. finally, the consistently convergent series development ^ \^^ Z_ r(p + i)iVv + p -:- A p=0 . V ' ^/ is given with- arg t, = if ^ ..is real and>0. /^.h An Estimate for Cylindrical Fimctions For tho cylindrical functions Z {t,) one obtedns by repeated application of (3. 13) (3.1^^) 12 MCA TM No. 1224 v^ ^ L'A \£) • ^ — > r-^ R H iNl fH • •\ • OO O m CO 1 cn • ""! • H CJ I + CM -:i- U rH O ^^ 1 -ri -p "- • ^ II q CM § CD U B H CD •P CB Ch + •g j ^^ I ^) o > — O X\ > CD P> ak-n <+H -P ^ ^ K ^^^ > u CD •H 1i oo O '^'^ — ^ O -s OJ . •^ '5 U O o.-n ^ § y — » "~H + H N — * CVJ + 1 p. ^ilCVi t-1 O H 1 CD 1 > N*^ l4H ,.-,> ^ O to u .,^ 1 . 1 tn CD f^ ^ H 1 • •H :? + C3 -P TT • CD ^--^ ^— y © © ^ ti ir\ — . -P 'H H + CM -* 1-^ H ^-n eg H ?• + -^ ,— ^ — , -^ H • S ^-/J(^ ■a-'JCJ J- •P m > -d Ch > rt iSj ti. ^— V •-• CU V, , 3 ra •H O H i_n H ,-^ jj Ci 1 1 P-l iS • 1 H 0' — » © ■P u o ?^ ?* H ,i^ ^ m -i-i + ■ U 1 o- ■ o O Ta ,_^ ^^ t^0^ H "h fH ^ > ^ © '.- ?- o ■A O tH 1 1 + 1 S o p C-i u fH ^ __?H CM =H C-, . o »\ + o- 1 K H u ■ B 1 > ^ s A" Oi\r-^ • -p 1 H , I + CD -P •^ 1 -— - ,_, ^ ?• Vn U H > ^ © •N 1 '-^ ' j^ W CO m + + r-\ m i_n ^-S 'jh p. vS © ■« V— ' CM -H ^-— ^ 1 ^ H f^ O + ^ !^ C-< M -p + H > C_i H tin * IS! „ U I ■f O OD II OJ r-. f^_ _^ p- U U fs "4 fn ^[■^i CD ,g Cj 'OJl i-n ^I -I- fi •H M © II CD •P iM © © .i^ ^— V ^ m ^ -P i_n o E^ ft o ^ o + :^ a D- rd •H C si O CO MCA TM m. 122h 13 is in every closed domain, exclvidlng the points and <» restricted (considered as a function of not depend on r . If Re V < 1 t ) } the upper limit does for such r 8 for which Ee is valid for negative r's. , this relation is valid at least (v + r) = 2. A corresponding estimate h. THE X-FIBICTIOWS OF THE FIRST AND GSCOriD ICEKD k.l Definition of the X~Functions of the Fjrst and Second Kind Since for y = the haaic equ.ation (2 J+g) is joined to the differential equation (3.I) of the general spherical functions, it suggests itself to develop the solutions o'f' the haslc equation in terms of spherical functions. One formulates the two at first formal series ^^'■hv,y) Z r=:~ 03 v,r V+r Ct-i) V \ iV (7)Q^ (I) r v-:-r (h.2) X'=- and attempts to determine the coefficients a (7) and the index v v,r in such a manner that these two series formally satisfy the "basic equation and converge . The further prohlem will he to investigate the convergence properties of the two series {k.l) and {k.2) in order to determine that, for the two series, one has to deal with analytic fimctions which, in general, are linearly independent solutions of the basic equation. H For the coefficients a (7) the indices v,ii v,r and the argument will he omitted where there is no danger of confusion; the same • applies to the coefficients to he introduced later for series developments of a similar kind. The summation index r assimies only even values. The term with r = in the two series (^.l) and (i+.2) is designated as the principal term of the series. In the solutions (^.1) and (^+.2) of the hasic eq^^ation an ai-hitrary constant factor remains 11+ NACA TM Ko. 122U •undetermined. It may be determined in some way. Hien the series {h .1) is denoted as X-f unction of the first kind and the series (^+.2) as X-fimction of the second kind irith the argument I and I'ith the indices V^i_l vith the parameter 7. It vill be foijnd that the index v is determined "by the separation parameter \} more acciirately, there exists a functional relation between \, v, l-i. and 7 which is expressed by X = A,^7) V {h.3) The series (4.1) and (^-.2) are now inserted in the basic equation^ the differential q-aotionts of the spherical functions are eliminated by means of the differential equation of the spherical functions (3«l), and the factor 6 of the spherical functions is eliminated by application of (3«3) • Then thei-e appears an infinite stim of spherical functions irlth coefficients independent of | which is equal to zero. The disappearaaice of the individual coefficient is sufficient to this end. This leads to the con6.itional equations -^,^^j, = cj^a^.g Prar+2 (r = 0, 12^ tl'. .) (^.ii) with the abbreviation &^ = -X -:- (V + r -:- l)(v + r) -!- 7- ^ Pt 2(v -1- r -'.- l)(v •!- r) - 2^^ - 1 (2V -!- 2r H- 3)(2V + 2r - 1) ^ (v + r + u + 2)(v + 1- H- |i + 1) (2v^- 2r -!■■ 5) (2^ + 2r -:- 3) (v + r - 1.-1) (v + r - !i - 1) (2V + 2r - 1) (2V •;- 2r - 3) (4.5) _>' MCA TM No. 122ij- " ' 15 i^-.2 General Qualities of the Coefficients a^ The recitrsion formiila for the coefficients a^ is interpreted as a difference equation. In order to avoid complications, the case of real fractional values of one-half for v is completely excluded and the case of real integers for v + IJ- and v " ^^ respectivelyj is postponed. Concerning the behavior of the coefficients a^ at infinity) a simple formulation can he obtained according to Kreuser (reference 12) . The equations lim Bup ^1 jal^ 2 (^^g^ X— > 00 \i r. 7! or lim Slip ,r.,tl: nr: = Hi. (k.^) are valid. If the behavior of the coefficients a^^ at inf'inity is given by {h.6) at least foi' negative or positive r's^, they increase too strongly to make a converr^ence of series (^-l) and ('+.2) possible. Tliersfore, a solution of the difference equation (k.h) is to be found vhich shows the behavior (4.7) for r—^ ro as well as for r — ?-», Althoiigh there always exists an esact solution which behaves for r~-^ - oo as indicated in (4.7); this solution will in general exhibit for r — >-co the behavior {k.6). Only for certain distinct valu.es of the iparameter V (free so far), the behavior ('■I-.7) prevails for both t—^°° and r— ^-co* inversely j, in this manner distinct values of X are coordinated to each valu.e of V . For 7=0 the conditTons are particTO-arly simple. There becomes for all r ' s X - (v + r + 1) (v -!■ r) a^ = (r = 0, t2^, ±h, . . .) Thus \ cBxi, for a £,iven V^ assume .any of the values (V + r + l) x (v + r) . It is determined by tiie reqviirement that the series C-i-.l) and (4.2) should be reduced to the principal term for this case, l6 MCA TM IJc. 122if Tjhich leads to \ = v (v -:- 1) . Ivow it is f vjrther required, that under X^ij) alvays the value should be understood ti>s.ch goes over to v(v -^ 1) for 7-$-0. The existence of s^lch a distinct X-value to each given V, [i, and e and its uniqueness will not he proved herej it folloirs from the method of calcxilation given in section 6 for the determination of X . From here on, the coefficients a (7) Trill alvays represent that solution of the difference equation {h.h) -i.-hich shovs -iiie ■behavior (h.^) for r — >t co^ belongs to the value X' (7), and therefore has the boimdary values lim a^. = (r = -2, th, t6, . . .) ('^.8) 7=0 Furthermore, the constant factor irhich is arbitrary in the coeff i' cients a^, may be determined in a given manner. U.3 Convergence of the Series Developments of the X-Ffnctions of the First and Second Kind From the estimates (3 •6) and (j-S) aB '■"ell as from the boundary values (i-i-.?) for r->l'K) there folloys immediately that the series (^J-.l) and ('i-.2) in each closed domain, i.'hich does not include the points i = -1, 00, irill converge absolutely and uniformly. One may fiu:'ther conclude that the series (U.l) and {h .2) v±H converge as -irell as the ejcponential series. Since the individual terms of these series are analytic functions in this d.omain, there follows from the uniform convergence that the sums of the series themselves TT-ill again be analytic functions, the singularities of vhich can lie only at ^ = ^"1,00, f rirthermore ^ that the series can be dif- ferentiated termT.Hse^ and there^ath the fact that the functions represented by these series are real solutions of th3 basic equation- MCA TM Wo. 122ll- 17 U.I; Further Solutions of the Basic Equation and Their Eelation to the X-Functions of the First and Second Kind Between the general spherical fimctions P*^, 1~^ } I'^ f V V '-v-l ^'^ > ^ } Q"'^> ^ > "^'^ ^l-"'- 0^ I'^hich satisfy the same ~-v-i V V -V-l -v-l differential equation there exist, in general, six linear relations independent of each other. They can he generalized for the X- functions of the first and second kind. To this end several relations for the coefficients a*"^ (7) ^rlll he derived. v,r The system of equations {h,k) and the system of equations originating from it hy the v<3uhstitution v— >-v-l and r— > -r are identical hecause of \x \i \x \x -V-l.-r v,r -V-l^~r v^r Due to the uniqueness of the solution there follows from it Furthermore, the constant factor irhich is arbitrary in the aj,'s can he determined in such a manner that a" , (r) = a^" (7) (U.IO) -v-l,-r v,r The system of equations {h,k) and the system of equations originating from it hy the suhstitution ij,— ^-|a hecome identical if one introdixces in the latter instead of the a '^ '3 the values v,r v,r^^^ r(v + r - ^ + 1) r(v + p. ■:- 1) v_,r'^^ l8 NACA TM No. 122ij- They are determined so that "bg = ag. Then the equations ■b^' (7) = a"^ (y) (1^.12) v,r v,r and x;^(7) = x^(7) ' (^.13) are valid. After these preparations, at firr^t a relation "between Z (|;7)r X^ ii}7), and X (7) is derived^ since according to (U .1) and (l^■.l2) -li(l) , , \ r (i , . -|a , . /, , , X |;7 = > i b (7)P (1) (U.lU r=-oo is valid. If one expresses in this equation the spherical f^mction P ""^ (I ) V+r hy P (1) and Q (O (under consideration of (ii-.l), (•+.^), v+r 'v-i-r ' and (^.11)) the required relation ^"'"'^'^' = H^-^tTi}['^f '(^'^) - r"' -^ - <'"te^^) vill "be found at once . In exactly the same tray there resuJLt the formulas (^.15) ^.l^i}7) -^^"^(^7) 0^-16) KACA TM No. 1224 19 ^fmm,^ ^ — ^ ^*-s. ^— ^ H f- 00 On o cd U P" H tA H OJ !-'. O 53 CO ■ ■ « • i-l •H -H H f^P> -P CO 03 H *--^ fH :^g PI Pi , II N (D •H "-.J E) ••s ^ -p y ^ i^ • S 3 O M iLfl -P f^ tti ^Q "* — ' O P! -H tH fH *■— -* ^ Ch O rd f-i -P ■rH H K^ O fi P) -< CO ^•*-^ -^-^^ 03 ■P i P! 1? /v 11 / ^^ 8 g& -p ^ s ^ 4JJ1 •H (U 0-j pi •^ fH / \?H K V_jr- t-) -P ?H r-\ > (D p- *- » a Cd +3 _, Ch +> OJ •H H -H fj O © u 03 '^— ' '^ (D rQ •H !/3 'd CO " — * O V iU fM in (0 P" CD !h K O (D trj Pi rt r3 fH •H »*N P-i fO ^ O H CD 4jLn •H ■H -P p 1 '-"^ i i -0 4J -'\ " CJ (D 1 nd S CO (i:> c:! l> O fl ta pl *— "^ K "^ — ■* fH o CO O fj CD -H cO CO "h H ^ — s. + ra rn -H .il i OJ •H o pi H ^ pi H t> + -r »— <• > o ci pi CO i ^- .> fl rt P" Pi l>5 CQ n d. X to ^ -d •H 03 -H H -P ?• ri •H -p P P' rQ rt 1 + ■H •H CQ •H S •-^ CD g pi s K m t--- o 03 O -H a > > ri > IH 4-= pi op: S *"~-^ •H + ■■ cvjI V:; H CD CT O -H ^ t-i C-, ra (D JS CC -■'-.' •H Pi P; Cy P^ O -H a •H 1 I O ■ O pi •rl ^-->. H • •H O P CO -ri CJi v; N II „^->. ffl P' ^

P! -P P> -H ?H © ^1 OJ 4iJ1 r—^ • ■% O -H © s 1 *» — ^ N *SJ\ -P ■-O >-5 t>^— V o 5 © (D ^-- H ••n '*^— ^ f^ O -H ^ CM O ^ c: 1 41/. y.^ ^ ffi iiO -P H fH © 53 II ^-^ ?^ I r-l Pi O o 'd o ■ • O ,Q ^ . = ^•^ — ^ ■ 4^ R) Fl !h ffl Pl^"- "Vh ^-^ W >. — s. pJ ci rt pQ e^ CJ T 1=; ^. O 4^ P! Pi ■•^ — ^ « "X > -p ffi OJ rQ Pi CO P^ **"^ <-^ [r- • O O CO Ch O OJ 'i ffi C -P -1. id^^-^ ° K- • d. + II O -H g ^ H l>i Pi <1^ H 't.^^ •H CO o > Pi H CO O -d M > .*— N, -P o -p •H • S -H fi II >- CO P^ 03 S J- -P •P *•> H '- — ^ 'r\ fH v_. +3 p., •H U S U/l (D CQ LTN^ .,-1 © pi ^ •H 1 fH -H H Eh ■^ •" _ P CO U m P » '-^ Pi M rd O H CO ^-^ H 3 «t/> v_-^ • H .el CQ H N ^-^^^ JJ 03 pi • P) © •H • •\ =y^ 03 fi cO^ ^ .Cl H txn -P q-l S o -P ^^^^ m o O -H H- M .^ — ^ CO •H P> -e! ,^ ,!=1 Pi C>J C\J H -P P" O P' P" ■iH > '^— ' h 03 g • -H •»v-H, r!:3 (3J + d. ?^ 0) CJ H 3 03 I- U- +^ M .C! P4 03 t|.| Pi •H > p> u !h © O I> OJ t>5 H CO H 03 ^1 !^ ffl CO nQ O- © O (C .CI O ■ri -P U © p! CO ^ CO H ■H •H Eh tH /\ to -P gi Hi 03 ^^ 03 £ P4 Pi b CT^H O fn •H CQ •H O © CO Pi O 20 MCA TM Wo. 122i^ tn -d- CJ CJ • ■ ,_^ ^ ^ ^ 3 OJ ■**— ' Nta^ CJ -—^ > • • OJ -=^ -=t- cu V— ' • CO © ^ © rd ^ O Is *^-N +3 cd -d H Pj fl P^ PI 3 © © Ujl •H ^•-^ ^•-^ UlA fH !> (•"^ C9 ^^.^ .-^ © CVJ :i> H © ■^— ^ o &i CJ PJ r-^ 1^^ •H • O H •P •H -d- cd Q « T^^ •H *=! , 3 r-J .g ^ ,r— ^ ^^ CJ r-3 • o OJ H H W ?H _ ■~^> m cd X +5 © a P! ^ P H P? o i • + o •H s_^ o e to 3; •H fn 1 •H P -P « a O Pi © t V ^ K •r^ ^A © *«*^ rt P4 Ch Pi -P V^ ?- p m '3 C-H © ^-~*^ ••\ pr. © l> ra d. 4U1 1 H + O © a X cd ,— ^ o S + ^■N h 4iLn ■;> >j O '— OJ >"-*• ^ to a PJ ^.=^ PJ •H m •H •H S o K • 'H fd to « ,E! ^ ^ '- '^ © © P! y^^. ^ +3 HICU HiCJ •S'^c^^ hTui HiCU H + ^ > , isD r-» § rd 1 © r^ « © .:t p! >> >^^-' V>.. ^>, 1 © O •H K fH ,0 r-s \i •H H II P! fH fH \=; II •P P Pi •H O + CO o ^ cd t> ,,r— V •H M © d .^"^ H •H CQ © > -H © •H CQ K HE! m • •\ fi M p^ CJ ^ •H p\ pS •H cd © -P >5 -H cd CM K H u/i •H ,Q -P O rJ cd CO 1 CQ H © ffl ^ pj ^> O rd g ^ 1 «;i ^ o 01 • — ^ Ph © S _. ti •H iiji O 'd r—^ /*^ o ■P M Cd © c^ CM^ c5 Cd rJ. > © H "^ 3 ^ Pi fn S ••s H P^ jjun *i, "^ m CO -P © O ^•— • M • fH •H •.-I ^ + fiH © "h H > CD p. ra ^'—^ g ^^ © © t>3 © a II G0> rS" ^ •H > *l/l o v_> .»— ^ H H i> f! S "^ ^> H f^i •rl S d o © pd cd +3 •«^ O A •H l+H ^ d ^ § h" 8 © un © ^ © ra -P ^ Ol 1 p! • ra -H ^ _^ •H g--' g a o © « r-H an h" ri > o r-^ fH -H (in <1 NACA TM Wo. 1224 21 These tvo general relations which are valid for |q > 1^ are in the case 1=1 tranGformed not to exactly (4.19) and (4.20); |e^^, namely^ represents in (4.23) and (4.24) an increase of the argument hy jr, whereby under certain conditions the hranch cut may he passed, whereas the argmient of -| in (4.19) and (4 .20) is ohtained hy choosing such a path from I to - i that the "branch cut extending from -«> to 1 will not he passed. 5. THE -Z-FUIJCTIOWS OF THE FffiST TO FOURTH KIW 5.1 Definition of the Z-Functions of the First to Fovrrth Kind If the two extra --essential sing'alarlties of the basic equa- tion (2.4g) are made to join, as indicated in section 2^ there originates, aside from an elementary transformation, Bessel's differential equation. It therefore suggests itself to attempt a solution of the basic equation also by series developments in terms of cylindrical f vinctions . The functions defined by the series (which are at first formal) ■zi;(^'(C;r) = (?' - /f r'' ^}_ ^'i^^ir) ♦„,,(?) (5.1) r=-oo oo Z^^'^(Or) = r?'-7')''''r^ "> ^>'' (7)n (0 (5.2) r=-oo are defined as E-f unctions of the first and second kind. In these '2 - -2 .2 ^ 2 series jarg ^1 < jt; arg k—I^ = o, if arg t; = arg y' Substitution of these series into the basic equation (2 .4g) (it is best to insert it into the transformed form (2 ,10) of the basic equation) , elimination of the first and second derivatives of the indices t , . and n , by means of (3. 10) and (3.11), and v+r v+r \ -> J \^ /} removal of the denominator ^ by means of (3.12) leads finally, exactly as in the X-functions of the first and second kind, to a three-term recursion system for the b' (7) . It agrees with the recursion system (4.4) for the t^ (7), if the li there is v,r 22 MCA TM Ko. I22U replaced "by -[i. The soli^.tion of the recTorsion sjstem differs from the indices t*^ (7) defined in (4.11) only "by a constant factor; V^r tliis factor is selected to equal one. Therefore the relation (U.3) formerly found "between the inde;: v of the generating functions in (5'1) and (5*2) and the separation parameter \ has to te assumed also in this case . As Z-f unctions of the third and fourth kind one defines ,^(3) .n(l) ^i(2) (f;;r) =<' 'if,?) + i^ (^^r) (.5.3) ^"v^^^(^;7) =2^v-'^(c;7) - i#v^'^c;7) (5.'^) They have the same relation to Eankel's fimctions as the -Z-f unctions of the first and second kind to Bessel's and Neumann's functions . 5-2 Convergence of the Series Developments of the Z-Functions of the ^'irst and Second Kind It must now he demonstrated that the series (5.I) and (5 '2) converge uniformly'- in a certain domain. One starts from the estimate (3 'IT) ai^<5. from the "boundary values (4.7) which are also valid for tlie "b . Tliere results v,r li2i sup ^r./To^ 7 . I ^' iZ " Ui (5.5) The convergence is uniform. Thus the series (5.I) and (5-2) con- verge uniformly and a"bsolutely in the entire domain \i\ >1 with the exclusion of the infinitely distant pomtj they represent HACA Tlyl ITo. 122ik 23 therefore analytic fvinctions^ can "be differentiated any nuinlDer of times tearmTri-se with respect to ^, and satisfy the basic equation. Only in special cases tliese series converge also for j|l =1. 5.3 General Relations hetween the S-Fimctions The transition to various f'onction tranches over the hranch cut from -"^ through to 1 is made possible by the general relations. They can be obtained corresponding to the case of the X-fijinctions from the general relations valid for the sepai-ate series terms, thus for the cylindrical functions. (Compare Magnus and Oberhettinger, elsewhere.) They read for j^] > [yj that is, | ?| > 1 2ii(l)(elni^.^) ^ gZV^ti 2^(l)(f^.y) (5.6) 2(i(2)(gZrti^.^) ^ g-2,(v+l)Tti z^(2)(^.y) + 2ie-'^^i/2 sin 2iT (v + ij cot (v + ^JnZ^^^^H^ }7) (5.?) sin (l - l)fv + iV Z^(3)(el«i^;^) = -e-^'ti/2 _i_^ 2^t(3) (^ .^) m (V + 2]'^ sin liv -!- -]-n ,-I.i/2 3-(v.l/2).i ^^ Z^^(^)(U7) (5.8) sin iv + ~\r. ^ sin (V -'■■ -irc sin (Z + l)(v + 3il jt -in (v ■ ^^ - " ^ sm V + ,:r It (5.9) 2k HACA TM Wo. 1221+ += O t I H P o •H ■P O & ^ \r\ H -p ir\ •H /-v H + O A_n m + •P as ts! q o 3 © O -P CO ^ II O CD H S l> O ffi H fc! ® O CD •H nci +2 O -P O H o ■p © o O CO -p •H © a •H O 'H Pl-P R>-P •H cd CO •p © O -P O -P •P U L_ /^^ P4 + > o CM © © f=^ -P O © CQ © •H tn I H ■d § © Pi O Eh V ■»_n W) V CM •d H H in CO H ITS « ^_^ o + Pk ^— ^ p 4^ >H o CM 1 + CM ft CQ si © •H •H CQ ^ © u CQ & y^ ■P O fl H © to © f^ ^ © 53 > -d G S O >5 H t:; © -P V ^ 4-n O CQ 60 ,0 U c5 (S3 © V ^ K CJ >5 I ,Q -d .\ © ,*— >^ rt c^ tH ••\ tH i.^ J fd ro © w^ 5-1 =^-,> «3 N .^ . ^ Si o Ch o & m •H -p CQ a © fH •H © O P4 •H & tH pi tH © © o rC! o +^ © 11 ^ n • ^ ,.— s CM **-** N V •♦^ A-r^ J>^ f^"^ M j- & i > tsl V *^ fH 1 O •— ' tH ^^ 'lAJ o MCA TM Wo. 1224 25 Tliis derivation is not accin-ate since the asymptotic develop- ments (5«10) are further dependent upon the condition |^| » |v + rl , and this condition is not satisfied for all series terms of equa- tions (5*1) aiid (5 '2), respectively, since the sum has to he formed over all r's from -00 to 00 . The fact that the developments (5 '12) are valid nevertheless is due to the behavior at infinity of the hj,'s (compare equation ('r.7)) according to which the aeries terms with sufficiently large values of r do not contrihu.te noticeahly to the Z-f unctions . Equation (5 'IS) is proved as follows. According to general theorems on the asymptotic hehavior of the solu.tions of homogeneous linear differential equations, the coefficients of which are polynomials, one obtains asymptotic series for the solutions by going into the dlf f ei-ential equation (2 .l+g) with a formulation of the form (5.12) and attempts to satisfy it formally. This yields for the present case for the coefficients C(p) the four- term recursion system (p + l)C(p ^- 1) + (p + l)p + 7'^ - X C(p) + I17 (n + p)C(p - 1) + ky^{\JL + p)(n + p - l)c(p - 2) = C(-l) = C(-2) =0, p = 0, 1, 2, 3, (5.IM from which they can be calculated recursively. This recursion system, however, is satisfied just then when the series (5 .13) are substituted for the coefficients C(p) . This su.bstitution leads after slight transformation to 26 MCA TM No. 1224 Wi H f O II ft 1 ^ 1 r-l 1 + OJ ^ K + + <— N r— > u f^ + + > > S..^,* /•*N .--*s H H + + fH h + + > > 1 ^ 1 1 ^-'. 1 ,,^ h" H 'ft 4- + -j- P< ft ^ + 1 ft cu ^ fH ^^ + -1- ^—^ ,,-^ r-J on > ?- **»^ ^'w-' 1 + C-, C-i ^^ , / P ft ' + 1 ^■-^^^ ^ u ?H %t a- + Zi^" > > .rO V™^ Jh C-H c ^ PJ , H O CQ Pi c^ ""Is CD > •H ^^— ^ O © A •\ > O fH •H ^ i > •H rt +5 ?H *» © j3 fiT fcD ?• 'ik CO o ~ ■fl fl © 'd 03 CQ a «H *-^ o ^ (D •r\ g A '■■p > (D >> *— ^ p{ 3 !h ?H H •H -E! ^ ft+^ ® rcJ CO © •> Pi © U +^ -— -H %% A CO H + -^ O • P tH H _ o Tli ti) ."— ^ ••v *---* ^ Ph (^ s^ y a N y.— ^ ?^ *» ■P ••>+' O a /Jl >» •H N—.^ ra tjD ••v ^S--" --^ (S Ch CQ + H ^ Ph ■P ^ ■sol H J_fi »^ CJ ^ ZL ft ^ ^ c; • N.-^ ©•— J>2 • +3 .s © f^A ^ o ZL > (D C.l'-^ i ^ '^ m -P A Ai • ri HH a •H -— '_ -J. 2 +. s.— ^ S ■P ijl ra O •S p! rf iH »v ^ -> !>5 •H fl CO -> O -d rH •P WH rQ ^ Ch O-- — ^ n CiH ti a © rQ +^ Ch -H i > o •H (V? A **— ^ •H + 'd +5 •H a? -p tq ^ CQ fj l>- _. rt -H § »s >-- o ca ■^g 3 • >Q PICO^ o "h ?H cy § •H , 4U1 1 ra Ch o ® a> •• -P t£lf\ ti Sh + rCj ^ o +2 O U >s«^ ti ft ©„H m -P -p o •A o 3 CQ g " ft •H +^ !> CQ -H A o Id f-' nd rQ 1 O CO H ^) O O *.— -, ■^ Pi Pi ?H tCl O (D o •H O \X) •H O 'H (» CQ fe; ^ +^ Pi • 1 Ch 0) O (D n -p .«•—*. a © M3 ffl ^ + i P^ ,i:l ® o OJ p; tjD ■»»-^ •\ ft > •P I S -P • CQ o ••- « CP ^1 d •H CO > o O Pi ra -H o ti t^ Cki O ^ ffl o fl fl O M ra > £ ■**,^ ,Q 1-1 > -d •H O •H © © O Pi ^ H ^n H p! O +5 cd ^ ■2 ft o o rt Ti O H o ,M "o Ci-i P^ HJ p © , ,o + CQ -H «i 1m O <*— V H O en •H Pi ?H ^ ■P o CB U H • © a cO Pi ra ft tlD (!) pi •H ^ • ITx Pi © -H C! fl rC! rH ■P ir\ v_^ .H Ch Cd f> .y o + •rH +3 o O OJ © iS O M •H i .H o CQ •P H ^1 rt a 4^ +5 ■P m as M ft • eo CQ 5 3 -d U •H ra «? H +3 o ^ ir\ •H -d •H o D3 rd f^ H + ft •H -H **— ^.'■•■^s ■H ■p C3 O Pi -H (D ?H O CO CQ H © ps -H © ^ **— -* -—+3 4J . CQ c ;::! Ki ^ © P > ^ 5 CM Pi -P P! in ©g ■P ^ 01 ^1 H P! © O O +3 O •H iH O ^ A ta ^ ^ ^ ^ S ir\ m C2 ft CQ +2 PI •P © Pi Ch P! ^-^ (S O O •H • •H A O >» •H fH tH -H © •tf -t^ 13 ■P .El A nd ,Q 'h to >3 ft U o P5 Pi 4^ ^ ra g © p: •P © -d fH Pi S *^-^ -d •H H !h DQ •H •H tlD PJ O CO o -d- g h tC ■d m ■P U CO Pi ra -H • ;3 !>s CQ © -P O CO +> • ^ CQ +5 a> ^ •^ ^ CQ ^ a o q ^»— ' •w o ,£J ^ 4^ • ^ PI Ih CD H ft 8 O ffl -t-5 -P ra CQ O © p P! •H p! P5 O rd S:3V • o +5 d-- -P © 5 !h Q .'-'N •H . o O fH cd CQ A •H © © 3 ft © N •N •P ■p •P E! ^ ro E^ 4J •P U -2 ^ CO (rf Pi +3 o O Ch CO t>5 O go© pi 8 CO H •^ CT rd H • 3 •H P 4^ CM ffl 1 CO H o cJ Pi (« ■d CM ra ra cfl • o ■H © H CO -P P • Ch .\ tsO LIA +3 O II l>5 CO • © Pi © CJ H fi a ffiv-' CJ S^ PI m © > •H o ■^ . fH ft !h w cd ^-- o •H + ,-x •n !>3 m ■p d © CQ += ,i:i ra O _ H o H "M o ^d !3D P! O M H •H P P • ft CQ -H rt a © K ffl oj In > ir\ II •H fS +■> 3 +3 rt © 'H S-H ^ + — ■P U ^, Cj P) P! © fH rt ft ■g EH pi h' H -H H O tij CO o ft ra cn H © ^ CQ o o ^ a U •H H +3 ^ -H M (D CQ « © Pi © © ■d fH O 'rt ]> P! £13 © ^ g:^ Pi A Ch P © Ch rQ O •H ^ (S o ■p o S ra MCA TM Wo. 122i|- 2? 5.5 Further Solutions of the Basic Equation and Their Eelation to the S-Functions of the First and Second Kind It is imiediately clear that with the Z-fuJictions zr) (^i7) and z'J^^^Or) the functions Z^^^^J^h'^;?) and Z~^^^-^'^'h^',7) also are solutions of the hasic equation. Since there exist only two linearily independent solutions of the "basic equation, it must be possible to express all solutions linearily by two of them. Because of the two relations * At,) = "cc^s V7tn it) - sin vjnl/ it) ! -v-1 V V y (5.16) n.v-i^^) = cos witf^(0 - sinwn^(0 J there follows with the aid of equations (^.12) and (-'-l-.lO) from the definitions (5.I) and (5.2) Z^^^\t^;y) = -BinvnZ}'^^U^',j) - ooBV^Z^^'^\i',y) (5.1?) -V-1 V V Z^^^^^;7) = cos VTi1^^'^\t,;7) - sin vj^''^^\^}7) (5.l3) -v-1 V V In order to e3q)ress the functions zj^^ '"'^ {i^',7) by Z^^ it,}?) and S (^^7), it will be practical to use the asymptotic series; it is sufficient to limit oneself to the first term of the series. Then thei-e becomes f'^'a;.)=ieK«-'f=")5 .V w^^'^^^i^li z V ■-■■■ ^ Z^ v,r b .19) 28 MCA TM No. I22U The only difference for the asymptotic series for Z^ ' ' {^fj) is that here a*^ (7) takes the place of "b (7) . There f ollovra ■ • v,r v,x' immediately that z"^^'^^;7) y iV (r) = z^^'^^r) y i^a^ (r) V ^ ^ — v,a" V ^ -^i-— v,r r=-co r=-oo (i = 1, 2) (5.20) By combination of equations (5. 17), (5«l8), ar-tcL (5 '20) finally, also the solutions 2,_lj_, (^^7) of the TDasic equation can te reduced to the two solutions Z ' (^/7)' As special cases of equations (5«17) and (5.18) -v-1 V (5.21) should he noted. 5*6 Laurent-Developments for X- and 5-Functions • The X- and Z-functions vere introduced vholly independent of each other. Since they all are, hovever, solutions of the same differential equation, it must he possible to express, for instance, the Z-fvinctions of the first to foiirth kind in general linearlly hy the X-f unctions of the first and second kind. It will appear that in general the Z-fimctions of the first and second kind are not proportional to the X-fuaictions of the first and the second kind, respectively. Thus, it is not possible to define simply fimctions of the first and second kind for the solutions of the basic equation; it must always be added whether one is dealing with X- or Z-functions. mCA TM Wo. 1224 29 CO a S ,11 IfN CiD H 'H ^ O «— O P4 P! • H O LIA O o in fl O — ' -H © CO > U g -P CM ^4oJ I CM •^ H HiCM II •H CM onlcJ + I + 01 /\li!. m CM I a I CM •k H I > 1 Al/l CM CM I > CM e _ ^4cM I ^CM H I CM CM H It ?- ■a-n H ' v— ?■ ^1 ■■H CM /\ CJ HCM tn ■«(Ln CJ l/^ I !> CJ > to 05. M -ci fl •H CJ o ^ WACA TM No. 122if 31 5.7 Connection "between the X- and -Z -Functions Due to the equality of the characteristic exponents V and -V - 1, respectively, in equations (5 -23) smd. (5 '2^) or equations (5-22) ar.d (5.25), respectively, X^^^' and ^^"'-^ on the one hand and X and ^ on the other on] y cilf f er every -V-1 V time by a factor independent of | . Thus one may equate -& it}7) = - sm (v - |a)7te K (7)X (I;/) (5-26) V -n V -v-1 u(l) (v-a)jri !a(2) u(2) 'Z (5;r) = e K (r)x (?;r) (5-27) -V-l V V The various factors, as sin (v - k^l^y and bo forth, were introduced for conve'nience. Between k (7) and k^^"^ {y) thei^e V V exists the connection i^(2)/ N 1 . / V -(|a+l)jti |i(l)/ , /^ „Q\ k;"^' ' iy) = i sin (V - lijire r. ' '(7) (5 •28) -V-1 ^ V One further ohtains with the aid of equations (^.l8) and (5 '17) V •-" ' V -(^l+v)Jtl sin_(v+LiW „^i(2). V ,^ V .29) cos V V . 2^^^hv,y) -^ Bin v^--J''^'-\i:,y) = e^^^^ k^^^'^7)X^^^^|;7) (5-30) If V, [1 are integers, those relations are essentially simplified; then the X-f imctions and the -Z-f -unctions of the first and the second Ifind, respectively, are actually proportional. 32 NACA TM No. 1224 Now the calculation of K (7) la left to "be performed. To that end one may select the coefficient of any pover of | in equations (6.22) and (5*25) and carry out the comparison. One oh tains »^v'"(r) 1 vn±, y\^-'^B B , = 2® f I) ir(i + v-M.-s) > iV (r) ^ — ■ v,r r=~o3 r - s 2 ^ M ■ s + r\"l 2 /J r=-oi r la i a (7) v,r "rfv . s - r 2 . |>(. . s + rN 2 /J (5.31) H(2)/ ^ 1 -(v+|i+l)jti/7\-2v-2s-2 Xi)" r(v + n + s + 1) > iV (7) r=-oo r(-v • r + s 2 t- 2/ •(- r - 2 n > r=-< i^al^ (7) s - r^ r(v.4-^-:-|)r(i. 2 (5.32) Any even nimber is to he s^ihstituted for s in equations (5 -Si) and (5-32) . The value of K\/''(y) is independent of the selected NACA TM Bo. 122i| 33 special value of s. If one replaces in equation (5 '32) [i by -[i and 8 ty -s and tlien multiplies hj equation (5*31), ti(l), , -H(2)/ ^ _ ^(^+l)TCi -1 S (r) k/^^^(7) =e^^--^^^-y- (5.33) is originated. 5 .8 Wronski's Determinant Wronski's determinant of the Z-f unctions of the first and second kind are defined hy % ^-^l^'-hUy) -%-<}^\Uy) -^'fhur) ■\^^,^'-\t;7) (5.3M -/& V dl ^ ^ d| '' From the haBic eqiiation (2 .kg) there f o3.1ows in the loioxni "VTay that Wronski's determinant of any two of its solutions is proportional to (l^ - l) ~ • The factor of proportionality is determined hy substituting their asymptotic series for the ■-Z-fuJictions of the first and second kind; it is sufficient to limit oneself to the first term (5 '19) • ' There results 7 6'- - 1 "^ — v,r (5-35) Wronski's determinant of the S-functions of the first and second kind results from T^ hy -asing equation (5*30) . 3h MCA TM No. 122^1 W,,Kf' WKf („W, (5.37) is originated and therefore 1 1 % = i^lD^^ (7) Lr=-co ^^^ % (r) \ ^^^ (5.38) Simplifications result for the inportant cpecial case la = 0. Firstj one agrees upon omitting the index \x when it has the value zero. NoT-r there is valid "b (7) = a (7) and further, according v,r v,r to equation (^.1), hecause of P.j^(l) = 1, X^^^(l;7) = y l^'a (7) V Z_ v,r r=-cQ (5-39) Thus, one can also T.rrlte for Wronslri's two determinants Wv = ii^^hi',y)i.^^\i.}7) e^-i = -7^% (5.i^0) 5*9 Other Series Developments of the Solu.tions of the Basic Equation Niven (reference 1) investigated series developments of the following fo3rffl (the functions represented "by them are called Y- and W-f unctions) : t(l) 1/2 -1 <^^^-;r) = O'^ H- 72)'/" w-^ 2_ ^^% r^'K^r^^^ ^5.ui) r=- 0; WACA TM No. 122i+ 35 00 ^J^h^nr) =y i'^d^' (7)* (v). (5A2) V Z — V i^ v+r r=-oo ' The relation of the variable \7 to I and ^ is: For the coefficients c-j^ and d^ there results again a three-term recureion system which can "be transformed into equation (^.^0 . If the coefficient of the principal term is set equal to a^, the equations p/v + r + (g ^ Ap/ v - n + 1\ r(iL±x_z_iL±_i)r(^4Ji + 1) [I / v^r d£ (0)/d| V / d"^ (r) = — — -^ f- sT (j) V;r p /y + r - u , Ap/V + u + 1\ v,r /. (0) = ir_vlr_a^^ (r) (5.1^5) are valid. 36 NACA TM Ko. 1224 The series (5-41) and {^.h2) convsrga -uniformly in each, closed domain given "bj \y\ < |-w) < co; or, expressed in the 5 -plane: 1< |vi^-l|<'»- The hotaiaang c-urve jyl^-l =1 is a lemniscate t Equation (5 •'+2) is, as irill be sho^ni later, a limiting case of a general development, which still contains an arbitrary parameter and which yields as a f\rrther limiting case the series (5«1) and (5*2) of the ^-functions. One can immediately give f-urther series developments of Y- and V7-f unctions; to this end one has to replace the functions ^y+,, in equations i^-hl) and (5.^2) by n^+y,, or the indices V + r by -V - r - 1, or |.i by -|a, or one has to make tira or three of these substitutions simiJ-taneously . One thus obtains a total of eight V-f unctions and eight W-f unctions. Their properties will not be Invostigated here mors closelyj it shoi^-ld only be mentioned that all of them also can be S3cpressed linearily by the -&- functions of the first and second ki^d which is done in the simplest iray witii the aid of the asyi^ijitc ''.ic series. Whereas the asymptotic series of the ^-fiTictions progress with powers of ^ , the asymptotic series of tl^e- '7~ end ¥-f unctions one obtains from equation (^^,41), and so for-tL, by substitution of tlie asymptotic series of the cylindrical fvnctions, contain powers /' p ry\-l/2 f o \-l''2 of \i;- - J'-) , that is, kI"- - 1) . According to a suggestion by Wilson (reference 7) one can now also set up asymptotic series which progress with powers of (? t l)" . They have compared with the series (5 -12) a sligi-^t '-idvantc-ge insofoi^ as a three-term recursion system results for their coeir'icients . Correspondingly, for the soltitions of the bsbic equation, also developments in terms of cylindrical functions i.dth the argoi-^ant b t 7 = 7(5 t 1) may be given, of the form 1 = (I - 1)^^/2(1 + l)-^/2 ^ e^ ^ (^ t y) (5.1^6) 1 t^„ v,tv+t •where t runs through all integers, the odd as well as tiie even ones . -^ , can again be replaced by n . , and so f ortla . v+t ^ V J y+^^ These developments will, however, not be followed up here . NACA TM Wo. 122it 37 6. CALCULATION OF TEE COEFFICIEKTS OF TEE SERIES DEVELOPMEKTS HT TERiyB OF SPHERICAL AKD CYLINDRICAL FUNCTIONS 6.1 Continued Fraction Developments The solution of the recursion sj'stem (i^•.U) which for r has the hehavior at infinity (^.7), can be represented "by the convergent (reference 13) continued fraction y\ A r-2 r\q^+2/^4+2l y^r+2!^. i 1 (6.1) The solution -■Thich has the 'behavior (^.7) for r represented hy the convergent continued fraction 00 can he % _ y\/^r\ _ y^-°^^r-2KA~2 I _ y\ -2^-r-jK-A-k ^2 (6.2) The suhnvjnerators of "both continued fractions are in each finite closed domain of y- and X-values for sufficiently large values of r in the case (6.1), of -r in the case (6.2) smaller than one-fourthj thus, according to a theorem on \miform convergence of continued fractions, the continued fractions (6.1) and (6.2), respectively, are in each doma-in of this kind for sufficiently large r's and -r's, respectively, uniformly, convergent and are therewith regulai^ analytic functions in y ' and \, since the individual approximation fractions are f -unctions of this kind. For not sufficiently large r's and -r's, respectively, then follows, that these continued fractions are also analytic functions which, however, need not in every case he regular. A solution of the recursion system {h.k) has now to he found which shows the behavior at infinity (^-7) for r — > <» as well as for r— > -«>. Tlien the valu.e of a-p/a;p_2 calculated from equation (6.1) must equal the value of this expression calculated 38 ■ • NACA TM No . 1224 from equation (6.2). An equation fe^aits which for given V and |i alloTTB calculation of the, separation parameter X as a function of 7. If 7 and X - v(v + l)' hoth are stiff iciently small, the solution X (7) is a regular analytic function of 7 which assumes for 7=0 the value v (v + 1). Thus the K^iy) as well as the e. Isi can he developed in power series in 7 with non- vanishing radius of convergence the magnitude of which will, not "be investigated here more closely . ' ■ '■ ' 6.2 Method for Mumerical Calculation of the Separation- - Parameter and the Development Coefficients The representation of the coefficient e^ by continued fractions is also for larger values of 7 still particularly su.itable for the numerical calculation of the separation parameter X and the a„'s. Mostly v, V-, and 7 are given. Then the values p^,, q^, and 0^ + X can he calculated nujuerically from equation (4.5). One starts from a value for X which is assumed as close as possible to the actual value and calculates for a selected fixed r the expression 3^+2/^ -from equation (6.1) as well as from equation (6.2), Then one repeats this calculation with a slightly altered value of X and examines whether thereby the agreement of the tiro values Sw.2/^r is improved. By further variation of X one can finally obtain .an,. agreement of arbitrary accuracy. Theremth one can find the value X (7) irLth any desired accuracy. One more investigation has to be made: whether the solution thus found for 7=0 goes over continuously into v(v + 1), that is, into X (O) and not perhaps into X (O); for X (O) ' V i- X V+2 ^ V+2 also is a solution of the present problem as can be recognized from the fact that equations (6.I) and (6.2) contain the values v and r only in the combination V + r. This question cannot be decided unless one has already a general picture of the functions X (7) as it is given in figure 1 for n = 0, v 's that are integers, and real 7 's. MCA TM No. 1224 39 The nvmiber of terms 'of the continued fractions (6.1) and (6.2) to "be included In the calculation corresponds to the desired accuracy . For large | r j the partial fractions y p q , o/^ ^-rM-2 assume the order of magnitude •y^/(l6r^); thus the index r' of the last partial fraction to be included ^ri-ll have to he selected at any rate larger than \y\j2. The calculation of the a (7) s is made by taking the v,r value found for X (7) as a "base, and calculating ^y,+2/^r ^^°™ equation (6.1) and therefrom a+2/ao^ ^hl^o> • • • 6.3 Power Series for Separation Parameter and Development Coefficients For the nianerical calculation of the separation parameter and the development coefficients one can for small values of I7I make good use of the poorer series developments in terms of 7 . If one limits oneself in -these to the first terms up to the fifth power of y'^, inclusive, one obtains, in general, still quite useful approxi- mations up to about 17^1 =5. Therefore, f ollov^ing, the power series for the X. (7) shall be calculated explicitly to 7 , o inclusive, for the ^/^q to 7 , inclusive. Therewith one more series term is obtained than by Wiven (reference l) } compared with Niven's cujnbersome treatment, the calculation is essentially simplified. For the limiting case 7=0 there follows from the recursion system (k.k) \^{y) + (v + r + l)(v + r) a^ = (6.3) The case where all a disappear is not of interest since it leads only to identically disappearing solutions of the basic equation. Thus there becomes A, (O) = v (v + l) , a„ = for r 7^ 0. V ^ ko NACA TM No. 122if For 7 » all 0^, with the exception of 0^ have nondisappearing limiting values. From the continued fraction (6.1) one can draw the conclusion a a r y Qr r-2 and from that further 1 + 0(/)] (r = 2, h, 6, . . .) (6.U) q_, (}, . . . q (— - (r = 2, ii-, 6, . . .) (6.5) If one takes the next partial fraction into consideration as veil, there results as the next approximstion = 7 r '^2 % • • • L'^-^i. . ^'-^6 1 + 7 17 v.- + V-p + . . ,. P-v>Qr-i-2 \ / An ^r'-'r-:-2/ (r = 2, 4, 6, . . .) {&^S) Accordingly, one oTo tains -r " ^-2P-'i ' ■ » P] ^ 0-20-1, • • • k '3r5r-2 "'K^. r5^r^2. 0(7^) (r :. -2,-1^,-6, . .. .) (6.7) One now substitutes ao/aQ- from equation {6.G) and ^-2/^0 from equation (6.7) into the equation r = of the recursion system (Jf-.U) and oId tains MCA TM No. 122ii- in 1> 'o ^0^2 r , u ^2^1^.^ ^oP- \ - 02 1 + 7 -.i-j- ■ ^2^k_ ■ U p 1+7 i^ M-2-^-1^ O 21". 1+ + 0(7^) (6.8) This eqii,at?".on peimlts the calculation of \ (7) as power inclusive. At first o series in terms of 7 ui^ to the power 7" , one can see, ty having 7 approach 0, that «o ~ 0(7)* Therewith, however, also is Imoi-m for any r vTith the exception of terms of the second and of higher powers in 7^ . If one now inserts $j. in -this approxiDiation on the right side of equation (6.8), c6 ■becomes already correct up to the third power in 7 , inclusive. If one repeats this procedure Trith the new values of the which are correct up to 7°, inclusive, there results finally 0^ and therewith \ (7) exactly up to 7-^'-', inclusive. The performance V of this calculation as well as the calculation of the a^'s is not particularly difficult, therefore the results are given immediately . In order to make the representation clearer, the following ahhreviations are introduced: = D^(l + 7^5,) +$, {^^9) where Dp = r (2 V + r + 1) (6.10) S-y. = 2(V^ - 1) (2v + 3)(2v - l)(2.v + 2r + 3) (2v + 2r - 1) •^ Do ' Pl,= _ -^2% Di, P.oQ 2^0 D-2 (6.11) k2 NACA TM No. 1224 h = ^2^2' ■" P-2^-2' Po '° c = ^i .. 'izisk Dr D -'2 ^-2 1 = 0, 1, 2, . . . y (6.12) Then tliere "becomes V vfv -!- 1^ + v^ gv + 2V - 2p. - 1 If V^V ' 1; + r ^2V + 3)(2v - 1) ^o^ ^ "^ "1^ P P. (AoB^ - A2 - 0)78 + A3 - 2A^Bi ■ B.^Ai ^ ~-(252 + 61,) P_o? (26 " ^-4j ylO + 0(712) (6.13) ^2 2 "^2 a. 6 |l - 7^52 H- /fs./ H- A__^ Vo TU V2^2 -■ ^Uj - ^2-^ ,.10 )■ H- 0(7^^) (6.14) — = 7 a ^2^4 1 - 7 ^ (^2 -^ &0 + 7 ^|-'2 + Ss^ -^ \ + -^^ 5];; — .10 + 0(7-^") (6.15) MCA TM Wo. 122l| i^3 r"(s2 + 5i+ -^ 55) + 0(710) (6.16) (6.17) V_o 2 i'-a .2, a^ 7^ ~ ^1 - r^s.2 + T'TS-g hfi 2 .._ ^-L" -^"^ D D -2 + 7^ - --^ ^ -^(25.2 + 6.4) - 5.2^ D-; D -2 J ao • + 0(7-^'^) (6.18) ^-k ^ ^k P-2P-i^ 1 - 7~(&-2 + ^-k) ^-2^-k 7 (5_2 ■'- 8.25_i^ + 8_i^' D -2 D-4 + 0(7!^^) (6.19) !i6 ^ ^6 P-2P-ifP- 6 ao D_2D.ij.D_g y% 2* ^-k* =-6 ;) ,.10 0(7-'-^) (6.20) a,8 . ?-2P-l,P-6P-8 , p. 10. D,2D_i^D.gD.3 (6.21) For the cane excluded above where v has fractional values of one-half, the convergence radii of these series equal zero. It seems therefore prohahle that the convergence radii are functions of v which can be infinitely large for sxDecial cases, "bu.t not in general. hh MCA TM Ho. 1221^- 6.h Power Series Developiaents Since occasionally power series developments of the solutions of the hasic equation (2 .kg) also can "be useful., they will be briefly discussed below. One can of course obtain them at once by substituting in the series (J+.l) and (^(-.2) for the X-f^tt^ctions of the first end second kind the Imo^m power series de-relopctsnts of the spherical functions in terms of powers of Ij one thtis obtains power series for the solutions of the basic equation which converge in the circle [|| < 1. The problem of the Laurent-series for 1 < III < <^ need not be discussed further since they are already calculated in equations (5.22) to (5.25). However, one can obtain these developments directly. Therewith a new method for the calculation of these functions and p., . particularly of X (7) is foimd. V One starts from the differential equation (2.10) which is written in terms of i rather than of ^ . (|2 -1) ^+2f| ^f) dl ^ ^^ d?. ^ ^ 2,2 -A. + 7 I u(u + 1) |2 V = (6.22) For tlae integration one tries the statement V s=-<» v-i-s (6.23) Then there results for the g_ (the indices \i and v as well as 3 the argument 7 in general are again omitted) the three -term recursion system MCA TM No. 122^1 k^ (v + s - u -t- 2)(v + 3 - la + l)gg+2 + |^(v + a + l)(v •:• b) - xjg^ - 7%.2 = (s = 0, +2, tk, . , .) {6.2k) There exists a solution with the "behavior at infinity > ■^ Sg_2 B a =■8 for a—>«> and a solution with the hehavior at infinity • feg-2 for s -» - «5 . The quotient of the two solutions is independent of s only then when X assumes certain distinct values . As one can see hy comparing with equation (5 '23), these are just the values \ (7). From the behavior at infinity of the coefficients g one can s conclude at once that the series (6.23) converges in the domain 1 < \i\ < » . If one substitutes in equation {6.2h) for the coefficients gg the coefficients calculated already'' in equation (5»23);there results after elementary transformations —: V;^ , J (2v + r + s - 1)(2 + r- 3) Kv +. r. + 1) (v + r) - \| -7^(v + s-)i-l)(v + e-|a)V=0 (s = 0, t2, tk, . . .) ■ (6.25) These relations can be used, like equation (5.I5), for the control of r ( u, numerically calculated values of the a (7) . h6 ITACA TM Ho. 122l| The recursion sj^stem (6. '2k) is, e:^cept for tho case of v^jj. being integers vith v ^ |j.| '^ 0, probaMy less suitable for the numerical calculation of the X ij) than the continued fractions (6.1) and (6.2) . Ordinary power series with increasing povrers of | result for the solutions of the basic equation if one sets equal g_p = g_. = . . . = and reqxiires g^ = 0. Then there results for v the determining equation (v - n)(v - 11 - 1) = (6.26) Therefore v has here a meaning different from the one it had so far. The behavior at infinity of the g^ for s — > -co is simple: all of them disappear. The behavj.or at infinity for s — > co is given by — >1 or — ^ p • The first case is the standard ^E-2 s case: the vo\rer series converges for [|j < 1. The second case is, for V and \x being integers with v = jfi) = 0, realized for a solution of the basic equation, the X-function of the first kind; the power series then converges for all finite I . It will be best to make the numerical calculation of the coefficients of these power series which are convergent in the M. , . imit circle so that first X (y) wj.ll be determined according V to the method given in section 6.1, or, for smaller values of 7, from the series (6tl3)j the coefficients g can then be calculated from equation (6 .'2.k) for each of the two v -values given by equation (6.26) . A special but simple problem will then be left: how the two calculated power series are connected iTith the X-functions of the first and second kind. MCA TM No. 1224 J+Y 7. ElGFHFlWCTIOnS OF TEE BASIC EQUATION 7«1 Limitation to v,|i Being Integers; v = Jm-I =0 The determining factors for the eigenvalu.es of the separation parameters X and p. end., if occasion arises, of the wave coef- ficient k, are the domain of space which was taken as a oasis, and the boimdary conditions on its boundary. Tliis treatise is limited to the most important type of eigenvalue problems of tliis kind; for them the domain of space lies eitlier within an ellipsoid of revolution, or between two conf ooal ellipsoids of revolution, or outside of an ellipsoid of revolution. The first two cases will be called problems of inside space, the last case problem of outside space. The entire domain -1 = r| = 1, ^ cp = 2jt becomes then effective for the two coordinates -r] and 9. Boundary conditions in T] and cp do not appear then; they are replaced by the requ-ire- ment that the wave f vinction for ' tj = I'l remains finite and that it ■ is single valued, that is, that it has the same value for q) + 2jt that it has for cp. The latter requirement leads to |i's that are integers, the first one to v's that are integers v = |ii| =0. That the X-functions of the first kind remain finite at the points Tl = +1 follows directly from the series (U.l) by taking the P^(|) = ^^ l^^'' i + \fl"2 - ij ^ ^hich i3 yalicL for estimate ""V this case, as a basis. v; Following, n will always be written for V and m for \i where V and |i are real integers; for the present, n ^ m & is assumed. The case of negative m's, the absolute amount of which is = n, is then obtained at once from equations (4.15), (^'17), and (5.20). Tlie calculation of these special fiuactions was practically settled amongst other things in the last sections; even though it was asstimed there that neither v h- ji nor v - p. are integers, almost all results can nevertheless be taken ovex- as simple limiting processes demonstrate. Only a few particularities resvLlt, compared with the general case; they will be discussed below. hB NACA TM Wo. 1224 7.2 Breeking Off of tho Series If q , = for a positive r' or p , = for a negative r', the a^'s "break off to the right or to the left, that is. for q,=0; r>0: a,=a,^=a,,=.,. .=0 r' ' r' r'+2 r'+4 for p^, = 0; r' < 0: a^^, = a^,_2 = a^,_^ = . . . = ,*> (7.1) is valid which follows in the simplest v?p.y from the continued fraction developments (6.1) and (6.2), Ihese cases occur" :7hen n - v is a positive integer or vhen |i + v is not a negative (Sic I) integer. Since it was presmiied = m = n the first possibility does not occtir, hut the second one does alwaj'-s occur, tiaat is, for all admissihle m,n. Here again two cases must he disting-aj.shed which are hoth originated from p t = 0: ■^r n + m+2 = -r'>0 or n + m+l = -r'>0 (7.2) In the first case m t n is an even numher, in the second, an odd numher; a 1 p is the first nonvanishing a . For the h 's there follows from equation (4.12) that they disappear for all r = r' + 2m. -,m Further, all P (^ ) disappear for n + r = -m, -m + 1, . . . m - 1. n+r The developments of the X- and ^f uiictions of the first kind hegin, therefore, for n - m = even with y .2 2^V2 .j3,r ^n,m-nV^) "^ V "1-^+2^2 (^) "^ • • • (7.3) MCA TM No. 122i^ h9 and for n - m = odd with m(l) , . m m-n+1 m , . + a' iffl- n,m"n+3 i^-^'-'-^pL.a) H- . . . m(l) , , / 2 ^Bi/2 -m -m+3 + t ,151 n,m-n+3 iii+3 ^..,(0 -^ . • • > (7.^) J The series for -Z (C;7) converges for all finite ^. The corresponding formulas for the -Z-functions of the second kind result if the functions ^ {t,) are replaced hy n (^ ) . The developments for the X-f unctions of the second kind shoi?- a special "behavior. , The , spherical functions of the second kind "belonging to the vanishing coefficients a^, i, a^i_2, • . • "become infinitely large in such a manner that their products have finite limiting values . The coefficients a (7) are defined "by m ^m lim a u . (5 ) = a P (^ j for m = 0, 1, 2^ , . . and v + r + m — > -1, -2^ . . . (7.5) Then there "becomes ^ / x' , /■ sin+n-i-l , , , , m , . a_, Ay) = lim (-1) r (1 + V + m)r(-v-m)a (7) for m = 0, 1, 2, . . , and v + r + m — ^ -1, -2, . . . (7.6) 50 NACA TM Wo. 122it and the series (ii-.2) reads r' .r m -.m n / n,r^'^'-n+r* ' r=-<» i'<,r(^'Cr«' (T-'" r=r '+2 The series (5 '^l) for v (w/y) breaks off only when n - m is an even number; for odd values of n - m the coefficients c (y) ' n,r" have the indefinite value oa.O if r = r' with finite limiting value. The series (5*^2) for w (v^y), on the other hand, breaks off only when n - m is an odd number; for even values of n - m the coefficients d (y) have the indefinite value oo.o with n,r ' finite limiting value if r = r . Similar conditions exist for the other V- and ¥-f unctions i 7*3 A Few Special Function Values From the series (5'1) one obtains when arg (^ - 1) = Jt f or ^ = fi W,o..,) J '^'"°''^"" '"'"-"'''/<"''") for n - m even ^ (7.8) for n - m odd J for n - m even i\f;fi^r/2)^bS^^.^^l(r)^r(|+m) for n-m odd r(7.9) MCA TM No. 122i)- 51 The X- and ^-functions of tlie first kind are for the index values n and m considered here either even or odd ftinctions of I or t,, resijectively, according to vhether n - m is an even or odd value. Fia-thermore , hecause of Q^(cos e + i.O) = (-i)^'^(coB 9) - iS- P'^Ccos 9 + i.O) there is n n 2 , ~n ^'='(o;r)=-i«i3^'^'(o,r) for n - m even (7.10) c<^^^(o;r) , <^'^o;r) u = -g^i dl for n - m odd (7.11) From Wronski ' s determinant (5 '35) follows m(l)/„. X n ^ ''' 7?r'^'io',r) d^ i^h^ .(r) r=-(» for n - m even (7-12) n a^ 00 "" > n, ,(r) _.^=~ 00 for n - m odd (7.13) Therefrom the -Z-function of the second kind and its derivative •with respect to ^ for ^ = can "be calculated at once . 7 'h Connection between the X- and -Z -Functions If V ,\i are integers, considerahle simplifications occur in the relations (H.I5) to (ij-.2o), (4.210, (^i-. 25), (5.6) to (5.9), (5.17), and (5.18). They are so ohvious that they need not he discussed further. Eqiiations (5.29) and (5. 30) now assume the simple form 52 NACA TM No. 122if m(i) , . m(i) , . m(i), . / . ^x n n n il-lh) For the k (y) sim-pler exoreasions can "be olDtained if s n in equations (5 'Si) and (5 '32) is selected in a suitable manner. The same expressions, however, result in an even simpler way if one substitutes in equation (j.lk) and in the derivative of this equation with respect to i, respectively, the special value | = 0. m(i) , . m(i) , . / . If one ejcpresses -2i„ (O;^) and d& {OfYj/di, respectively, according to equations (T«S)^ (7 '9), (T«12), (7 '13), using equation (5. 13), there originates for n - m = even , m m(l) 1 1/2 m/yVa ^n.m-n^^) K (7) = 2^ 1 [IT (1) ' .3 X n m{2) •«„ (7) = n ■" ' (2; ^.,x;^^^^(o;.)r(|-m) m n,-n-m (7) h (7.15) and for n - m = odd m(l), . K (7) = i«^/^ i^/z)"'-'^ ^n,m-n+l^ ^^ d2^^1^0;7)/d| r(| + m) ^ (7.16) n ^'^ -1/2 -m/y\m-2 ax;^^^^(o;r)/d|r(|-m) m . / X a -. (7) n,-n-ijti-l^' "^ MCA TM Wo. I22U 53 By f^^-'-^COjr) and clf^^'''^ (0;r)/ai the values of these functions are understood which result vh^en i goes to'.rards zero from the positive imaginary half plane . The distinction between ®T«n aad odd n - m can he avoided if one sets, for instance, s equal zero in calculating the « (7) from equations (5. 31) and (5'32)j the formulas (7»15) and (l .16) , on the other hand, have the advantage of greater simplicity. 7.5 Normalization and Properties of Orthogonality of the X-Functions of the First Kind The eigenvalues of the hasie equation l^iy) are alvrays real. Proof of it is given in the knoTOi manner. Equally simply it can he m(l), sho-vjn that the functions X^^^ (1^7) ai'e orthogonal to each other, that is. 3C^^^(i;r)#^(e;7) ^i =o (7.17) n "" ' n 1-1 is valid for n j^ n ' . By inserting the series (k.l) into (7 •17) one can also express this property of orthogonality for even differences n - n' thus: \~ a"" (7)h"', _ ,(7) o-irl-^-T = for n ^ n' / n,r" n ,r+n-n'^''' 2n + 2r + 1 ^ (7.18) r=- 00 For the normalization integral one obtains ,.(1)4.^,^(1) J.,) « = {^Y SrTlFTT<,r(^>^n,r(^' f^-^') 3pr:-oo 3h MCA TM No. 122i^ 7.6 Generalization of F. E'. Nettmann's Integral delation In tlie case m = one obtains a second solution of the "basic equation {2. kg) which is independent of X^~^ {i}'/) in the form of the intesral F(0=i/ efl^li(i)(t;r) 2 / _^ I - t n dt (7.20) The fact that this integral actually represents a solution of equation (2 J+g) is confirmed hy suhstlttition. The calculation is reproduced in detail in Bouvkamp (reference 10) • For large |, t and i in the denominator of the integrand cancel in first approximation and one can see then at once that J(0 is proportional to the -g-fxmction of the f oirrth kind. The integral over t can then "be evaluated according to equations (8 .2o) and there originates, because of equation (5. 12), J(|) = -i7 z^hn}i) -iz^^^'(7i;i) F(7;7)/V i -^^'^7 -,7 T m a : (7) n,r ' (7.21) r=- 00 According to equations (7*8) and (5«33) the -Z-f imctions are now converted to X-f unctions . Because of equation (5 '39) there results finally x[f^i;7) 17 n 4^^U7) 1 2 ni ^-prT-Aj^-^^(t;7) at (7.22) Therefrom results for 7 = F. E. Keiimann's integral relation betveen spherical functions of the first and second kind. MACA TM No. 1224 55 7.7 Zeros of the Eigenf unctions For m > the zeros of the "basic equation are situated at I = tl, respectively, sixice they there have the "behavior (^ ij • If one divides the eigenf unctions "by this expression, the quotient does not have zeros at I = tl. In order to understand 'this, one need only enter the "basic equation (2.i|-g) vith the expression (I t l)''^' " multplied "by a power series in (I t 1). The zeros of (^^- - l)""^^ X^ (|;7) are all eimplej if they were not simple, all higher derivatives of the eigenf unctions would have to disappear there also. Since it is, however, a non- identically vanishing anal;'-tic f-jnction, this case can never occur. Further properties of the zeros of the eigenf unctions follow from a simple consideration of continuity: namely, that in a nonsingular point of the "basic equation a zero cannot "be newly originated for a change of . . ."* and an already existing one cannot vanish. There- with the problem of the nvunber of the zeros is essentially reduced to the problem of the number of zeros of Legendre's and their associated polynoDiials and of Bessel's functions with on index of a fractional value of one-half . One deals first with the a (I;^) m'-th real | and 7, that is, with the eigenf unctions of the prolate ellipsoid of revolution. All zeros are real; for this is valid for 7=0. If, namely, for a change of 7 a complex zero would originate, the conjugate-complex would originate along \rith itj but it contradicts the simplicity of the zeros, that a real zero splits into two complex zeros. The number of the zeros in the interval -1 < I <+ 1 equals n - m, that is, the number of zeros of P (|) in this interval. The zeros outside of this interval go over into the zeros of Jn+l/2^^^^ •^'^■'^ ^ — ^ ^^ "''^® asymptotic distribution of the zeros for large I is the same as' the distribution of J .•> /niy^) for arbitrary 7 . For the eigenf unctions of the oblate ellipsoid of revo- lution A (Iji7) .with real 5 and 7 also n - m zeros are situated in the interval -1 < | < 1; but now the remaining zeros Translator's note: . . . missing in the original. 56 MCA TM No. 122lf en OJ © © ^ ^ (0 01 t:* cr © ■. P4 ^~~ p >> ■H § -d >= -d- CO ;>= (D -p H ^ H fi ' -d (>J • HJ H +3 c - M.^ t>- CO •H .-— ^ S v-- (u ■H ••\ M N o u _, ® pi © ^—^ 'aP' ^t P, CQ Pi § •p. ' ^ £=• O © OQ •* © E •H d CQ H !h S^rt 1 u i -p 3 CQ Pi •H p- 03 N Ps cj o © 1 5 »»\ ■P ^ H ffi • O ,Q P" tM p '^ P< H'-v •H iVh 'cu^ O ^ ■^'^ Ph^jji -P ^ p- 03 0^-^ -P Pi a K Q © H f! -ci •ii o O -H fi CQ 1 •H o f-- © X •H ^ § ^ H H P ^^ -p -. S-P ? © "^ •H V_--' •H CO l>5 CO u3 pi ^ H © +3 ^— ^ -P P" ;::; H ^ u cy -^^ ± t!D {=• 'g •U) cr © fH -rH © ri •H rt f^ © > O C'-l liH "-a M 3 S~^Pl a •H ■H O H O © 1 Ph © H -P O © CO rO ra © cr !h <- ' ^^ 1 Si © + ^ •H ra S +3 It fl fH CQ •H S o ch H H rt ra •H © Pi g:^ CQ ^ r « +3 o"^ ^■H o o CO -H H -P CQ § o J !>s -^ r?^ ■H II rQ CQ U/l © -H Pi Q Pl -P a © J>v f! ^ © P^ g-^ 1 o ^•— s. h •H fH P Pi •H N rt ^•-^ H -P IM •H 'H ■P N».> © H CQ © 1 OD rj P! ?H CQ fH fl M • i ^ ;i ■^ to 9^ ^ .^ ^ -H o J'— » ^ H CT © ^ hJ O tH © <>«. © •H 11 dj © S o © Ch a4 -p ^ 1 a a Pi © l-l +5 r— K aj © ?v H _ N H 1 © ,^ © CO g Pi *\ ?^ UO+3 >;, •H +3 fn 2 ^ • •- • © l-i Q • ^i "a Pi fH ■P !h ■ fl ^X)■'^ ^ — . ^— «.* fn CU a 'H Sd CO ^,^— ^^^ CO fl r ^■ Pi pi _, «3 t- ^ ^ © H o © H -ri +3 _ CQ -H H -P •H CO §!l gg . a ^ H H 1 =H -P q --- fH O g £ +.-> Q j: U «i CM 5-" 3 (^J Tl, ^^ C! V ai/i. P- Ch Ch tiO > ;^ fH ^— ^ OJ ^H © •H •H CQ Ph txn © ^ ■ »- • • 1 1 O O © -P Rj •^M- N !> *'— N t- Cfl ^ O •H O '5" g H +^ © m f>s 53 ^"'^ s " to ^ O H +5 m ^ -U 1 ri fl p- Ch a +3 P4i- CQ a © 3 o ro uji O •H (0 Q O ■p CQ (U fl •H N 4J Ch © o a !h •H -p 1 •ri Pi O O ■^ "^ tH ^ CO W © O © -p +> © ■:^ s !^ •H fl © _ o ^1 •H o-i © -P Q H f^ -s © cy H ,^ •-^ y O O & t> rt C3 -p © g o !h © ^ •H 8 ^ u fH Ch! o © •H CO 3 co the development (8.8) is reduced to the principal term with' the N real zeros of Esmjite's H*^ polynomial, Trhere as the X-function with the indices n and m to "be approximated has exactly n - m real zeros in the JXiterval -1 < | < 1. Therefrom follows H = n - m. For negative m one inserts instead W = n + m. 8.3 Asymptotic Behavior of the Eigenf unctions for Large Eeal 7 The . asymptotic representation of the eigenf unctions results lay calculation of the coefficients 13^,. They read, aside from the terms of the order y -^ , 62 WACA 'TM No. 122i|- '-2 ■a, m. i)-7 _ m -U 1 ^tm2 327 (]\r - 25E - 36]^ 47 1 + _L.(ii-2 + 2711 - 10) 327 H(TT - 1) 1 327 1 + i/N + S - Sm^") 27V 2 y > (8.11) 1 327 1 + l.(li - i + 2m^) 27 V 2 / -6 m m ^0 i^^f ^0 1287^ liT'eyT N(K - 1) (N - 2) (IT - 3) n: '8 1 -8 Nl ..2' ■e^ '^OUQy^' ^^ 20U87 2 (N - 8) I According to the type of derivation, however, the eigenf unctions are approximated hy these series only in the interval -1 < | < l. In order to oh tain an asymptotic series also for other | one starts from the integral equation (7 '2k) and substitutes for the X-f unctions in the integrand the series (8.8). Therewith the asymptotic develop- ment of the eigenfunctions for all | is toiovn; in particular, their hehavior can he investigated where, besides 7, I aJ.so is very large. Since now the eigenf unction for all p is asymptotically kno'i'm.^ one obtains the solutions of the second kind by calculating the integral in equation il '2k) -vrith the asymptotic series of the eigen- f unction and by means of another appropriate path of integration. MCA TM No. 122i^ .'63 The zeros of the eigenftinctione located in the interval -1 < I < 1 crowd for large 7 more and more aroiind I = Oj in order to •understand this, one has only to divide the zeros of Hermite's N'*'" polynomial "by \f^ and therevrith to convert to the i -scale . The domain of validity for equation (8 .10) and (8 .11) extends over the indicated domain; thus originates; for instance, for m = ti from equation (8.10) the asymptotic representation of the eigenvalues of Mathieu's differential eqtiation foi;nd by Ince (reference ih) . However, the limits for this domajji of validity shall not he submitted to closer investigation here . 8 .U Asymptotic Behavior of the Eigenvalues for Large Piirely Imaginary 7 One limits oneself again to n and m that are integers (n = m = 0) and to .purely imaginary 7 of large absolute value . This procedure yields the asymptotic s of the eigenvalues aiad eigen- functions for the coordinates of the oblate ellipsoid of revoliition (reference 10) and for the so-called inner equation for the separation of the wave equation of the ion of the hydrogen molecule (reference 7) • The method applied in equation (8.2) fails here; 7| namely would become purely imaginary and the DjjA\/ 27 \ would, for large i, no longer decrease exponentially, but increase exponentially; they would, therefore, be no longer appropriate for the development of the eigenf unctions . The wave mechanical picture of the differential equation (8.5) shows that in the case of purely imaginary 7 two domains with low potential energy are present at 0=0 and Q = 2n, which are separated by a high potential pealc with the maximum at = ^-^ One may, therefore, expect beforehand that the eigenvalues will degenerate in first approxijaation; their split-up is exponentially small in 1 7 j; it is the larger, the higher the eigenvalue. For each •eigenvalue there is an eigenfunction symmetric with respect to e = '^, that is, 1=0 and an asymmetric eigenf imction . The mathematical treatment is as follows . A singularity is made to move to infinity. Then one obtains from equation (S.J+g), aside from an elementary transformation, the differential equation of 6h NACA TM No. 122A Laguerre's orthogonal polynomials. This suggests for the solution of eq.uation (2.%) the formulation of Svartholm (reference 7) Fl(?) = (l - 1^) e" 2_ a^%^.t|2p(l - U] (8.12) t=-oo wherein 17 was set equal to pj again it does not mean an essential restriction if p > is assumed. By substitution of equation (8.12) into the differential equation 2.4g), application of Laguerre's differential equation, and the reciursion formiilas for Laguerre's polynomials (compare Magnus and Oberhett inger (reference 11)), -there originates in the known way for the at a three— term recursion system. With the abbreviations X = -p2 + 27P - hr^ + 1 - m^) + A (8.13) T = 2H + m + 1 (8.14) 4At = (t + 2t - 1) - m^; P^ = 2t(T - 2p + t) (8.I5) the recursion system reads ^t+1 -Shi + Vi h = ^^ + ^t)<^t ("^ = -'^'^ -^^^+-L^ -^ + 2, . . . (8.16) Therefrom follows for A the transcendent equation A = 1 - I ■' . . . + 1 - f ■ - . . . \cS.i7) A + P2_ JA + Pg A + P.i Uv + P.2 from which A can be obtained as series in terms of powers of p Therefrom then results -1 HACA TM no. 1224 65 X^(7) = -p2 + 2tp - |(t2 + 1 - ni2) - J-(t2 + 1 - m2) 6l4.p' • J^tI^ + 10r2 + 1 - 22i2(3t2 + 1) + m^ —^Rst^ + Ukr^ + 3T-2ir^(23T2+25)+13m^^] + o(|7r^) (8.IS) 5l2p^L - -i ^ 8.5 Asymptotic Behavior of the Eigenf unctions for Large Pvirely Imaginary 7 For the coefficients of the development (8.12) <^0 !l2 -j^^..l)^-.g(l.4^).o(H-3) _!_[(,., )2_2]^(,, 3)2 _2], 0(1,, -3) ife[^-^)^--'](-^p^)^l + 1 for odd n - m; thus 66 MCA TM No. 122i^ T =n+ l = 2lT + m+ 1 for n - m = even = n = 2TJ + m + 1 1 for n - m = odd (8 .20) is valid. BalDer and Easse (reference 7) calculated the series (8 .18) "VTith the exception of the last two tenns; onl.y for the special case N = they give also the last two terms^ Bouwkajnp (reference 10) calculated the series (8.l8) \r±th the exception of the last term for the special case m = 0. The asymptotic series (8 .l8) can still he used for m = ipj it then goes over, exactl;;^ like equation (3.10), into the asymptotic series for the eigenvalues of Mathieu's functions (reference ik) . . For large valiies of \y\ the eigenvalues move closer and closer together in pairs so that the asymptotic series (3 .18) for the eigen- values of each pair axe the same (see equations (8.20)); that is, the difference of the two eigenvalues has a stronger tendency to vanish with increasing jyj than any power of l/} 7 j . (Compare tahle 11.) The series (8.12) for the eigenf unction is useless in the interval -1 s | ^ . There an approximation maist he attempted starting from the point I = -1. Since tlie eigenf unctions hecome exponentially small in the neighhorliood of 1=0, one can huild up the eigsnfunction in the entire interval -1 < I < 1 "by comhi- nation of the two approximations starting frcaa -1 and 1 and one obtains ^J^Ui;y) - constant (l - 1^^'^- J oAj^ -^^%^(^ - 5)1 t=-CO L t e"^^ L^^^r-2p(l-0 '■ (8.21) For even n - m the positive, for odd n - m the negative sign is to he selected; in the one case the eigenf unction is symmetric, in the other antisymmetric with respect to "the point | = . MCA TM Wo. 1224 67 'What was said in eection 8.3 is valid for the asymptotic calcTjlatlon of the eigenfujictions and the fvmctions of the second kind for any conrplex | as veil as for the limits of the domain of validity in the variables v,|j,,7 of the asymptotic representations. In order to show the use of the asymptotic series for numerical purposes one compares for m = a few eigenvalues with the values resulting from equations (8.10) aixd (8.I8) by giving the value of the remainder term 0(7"3) and 0(p"^), respectively.- n 2 ,2 10 -25 -100 -100 \(r) 2.305 -16.07904 -81.02794 -45 .48967 Eemainder tez-m -0 .025 -0 .01616 -0.00008 -0 .01528 9 . EIGENFUIICTIOKS OF THE WAVE EQUATION IW EOTATIONAIiY SYMT/IETRICAL ELLIPTIC COOKDHTATES 9.1 Lame's Wave Fvmctions of the Prolate Ellipsoid of Eevolution By separation of the wave equation in the coordinates of the prolate ellipsoid of revolution one obtains the following solutions of the wave eqimtion u = P^^^^\}y) +Bz'^J^^(^i7)] C#/^^(r,;7) -K^^U,;y) Ee^^^ + Ee- V i|icp\ (9.1) A, B, C, D, E, F are arbitrary constants, v ani vl arbitrary real or complex parameters; the significance of y is given by equation (2.8), thus y is real. The coordinates | and ^ = 7I , respectively, r; and cp are real as well. Under X*^^ ' i'^iy) one 68 ItlCA TM No. 1224 /^'' understands xT) ' (t + i x Ojy) . According to the kind of the boundary value problem presented, the arbitrariness concerning the constants and parameters is limited; then such solutions of the wave function have to be determined •which remain finite for the entire domain of the eigenf unctions . Folloving^ as before, three-dimensional domains only are dealt vith which lie inside or oti.tside of an ellipsoid of revolution or betireen two confocal ellipsoids of revolution. Then the domain of the coordinates t) and cp is given by -1 = ^=1, = cp = 2rt . !Ehe requirement of b ingle -valuedness and f initeness of the eigen- f unctions then leads t7 v = n, |i = m, n ^ jmj ^ 0, and D = 0. The eigenfunctions are VTritten in -the form u„(t,n,cp;k) = n <^^^a;7) ^ <''^a;7) 2^^-^(ri;7)e^ (n = 0, 1, 2, . . . ; m = 0, t±, t2 , , , ., tn) (9.2) The domain of variables in | is denoted by |tl'^I"^I2 and ^.j^ ^ ^ ^ t,2, respectively. ■ For the prolate ellipsoid of revolution there is always 1 '^ |, . For inside space problems §2 = finite, for outside space problems infinite. For inside space problems boundary conditions for |-]_ and |p are to be presci-ibed. This results in two linear homogeneous determining equations for A and B; they can be satisfied only for certain distinct A^alu-es of y, that is, for certain eigenfrequencies; in that case they fix the ratio A:B. In case ^i = 1 ^ boundary condition can be prescribed only for Ip > Ij "^^^ boimdary condition for ^1 = 1 is then replaced by the requirement of f initeness of the eigenf unction at the singular point i± = 1} it leads to B = 0. For outside space problems the boundary condition for I2 = " is eliirJLnated; the functions (9*2) have for I2 — >• " ^°^ arbitrary A and B an oscillating behavior . One can see that immediately from the asymptotic series (5 .12). The boundary condition at | = |-j_ gives the ratio A:B. For t-^. = ^ this boxmdary condition in tvm is MCA TM llo. 122il- 69 eliminated and B 136001168 B - 0. A condition for the frequence does not exist; all wave coefficients are ac'mlssible^ the spectrum is continuous and extends from k = to k = «>. 9.2 Lame's Wave Functions of the Oblate Ellipsoid of Revolution The solutions of the T^rave equation originating "by separation of the -wave equation in the coordinates of the olDlate ellipsoid of revolution are obtained from equation (9«l), "by replacing y there by tiy. Here also only the three -d^jaensional domains characterized in section 9.1 are dealt with and the eigenfunctions can, therefore, "be written in the form (n = 0, 1, 2, . . . j m = 0, tl, ±2, . . . , ±n) (9-3) The domain of variables in r| and cp is the same as in the coordinates of the prolate ellipsoid of revolution. The domain of variables in ^ is again denoted by ^-j^ = ^ = ^2 • ^o^r the oblate ellipsoid of revolution there is = ^-|_. What was said in section 9-1 for |, > 1 is valid for inside and outside space problems with ^1 > 0« However, whereas there |-j_ = 1 was a singular point of the -&-function of the second kind, here ^ = is a regiilar point for all ■&- functions . Tlius, for determination of the eigenvalue problem for {!]_ = in this case, also a boundary condition must be given. The area ^-^ = is a circular disc. If such a circular disc actually exists as a physical object, for instance, a circular screen for problems of diffraction or a circuJLar membrane, the boundary condition on the disc resuJ.ts from the physical problem taken as a basis. If, however, this circular disc has geometrical significance only as singular surface of the coordir-ate system taken as a basis, for instance, for the determination of the acoustic or electrical natural oscillations inside an obD.ate ellipsoid of revolution^ the eigenf unction together i-Tltlz its derivative must be required to be continuous at this circular disc which leads to B = 0. TO MCA TM No. 122il. a J- irs o\ vo I © o Pi •H CQ o s •H ■P O I CO O F! O •H +5 •H S (D Q O -H -P n:! (D >a O 'H © Pi d o •H o ^ £h q m 3 -P g ■^ o •H i ^ 2 +> d bU O © •H H PJ O 3 © '© © t! © ^ :^ ^ ^ Hi •H ^M © ri ^ O © •H -P g ^ ra •H -P ^ •a •H crt t)l) O ^ ^ Co ra H O ^ © m © O M f:^ xi •P O C) n •p CO fe m rt ^a O") -p H Pi • ■H © On a m ti •H • o H a © n -p -p © •H O fn a ■p .a^ o pi H -d ci o Sh © > O P 5 ON © ^ ^^ ^•^ 0!) fii ■P i^ a ^ a fl O 'ti %^ ■H r-l •^ +3 pi fH O O © d > ;> 3 © 1 © • 'z"-"^' a ^ 2 ^ ago m o •H 't;.^ •H ©„ 8 w ra Mm tH ro/TD m •H O © © © • o >s pi ^ M f^ +5 © 43 .^4i Pi -P 1 pi er © S -S 05 © A3/^ M ^ -d © ^^K ■P •H -* H vL^ i^ iM H © i^ O O g ^ U +3 OJ o pi -P p- 1 cr ra m o a ® fi ra fl 1 -p ■H © •H 05 ■d 05 o OS , H ^ 3 p o © ■P 1 -^ 1 ?3 •o 3 o fr g Is o P s /D rv a H -P ra + ra ^^ ■H o © ^ ra © o -P H •H ■p ch H a >D/r) o cC fi Cj J3 H •H pi pj •H © o ti Q r^. bD fH 1 •H © +3 ^ii o s ■P :^ . ^ o pi un c 4J o /T5 /O © fH f^ o Ch © O o tH o •H +3 * ^v!^ Oi ti o Pi ^x^ © O •H H rf fi ^ -H ra H -P O +5 -P © H 1 •H O Oj •H © ra © ■S'^fe -P e! OJ 1 "^ , ■H N +3 © © . 'H 1 ^ \ tH •H O +2 Pi •H H Pi fl o rj tUl 6, C3 -H S "d O jii O ^ •H /TD/T5 •H ra © ■P II S P ra ^ pi * tH^g © © h O^^ !> ■P OJ ITS U © o M ^ • © tH > m •H O C7N -P o ^ U tH © as 'd o © 1 •P m "^^M o $ cu © •P © Ch Pi IjM g © © -d 1 EACA TM No. 1222f *— » t^ c^ • IS ,C1 ^ ct H ■P !> U>1 401 -d U a a a (T U -ri U -Ci ^ t)0 ra M (DO) O H H •P -P -P r 1 2 P! fn P! — 3 a -H (D -H CO 1 [^ 1 ):: -^ ^.co^ j CD 11 Xi H t ■p d -P pi pi 60 K ra o fri fi OJ (D •H t' 60 S tQ tjO-H 0) (D >a r O "^ t3D • 3 pr o c^ o •a rtvo^ fH ^^-s o • rt o .3 OJ •H o^ o +3 p- ■g-:;^^ t>5 1 B g S ;zj cu Vi -H a" • d U/l +5 o o fj >—• ^S<*H .. H H gi c o CO •p r- 1 ~:3 © © © R) — ' © -P ;cj S^. fl 1 rQ -H ^ © P 1 rt -p o •H % s d CO It -d a H d © © 03 U d o © .+3 -p W w fd ra H d A ra H ~4 «/l ^-1 vu >5 O © + >j © ^ -e tH o O PM CO 0> H (M CM k^Jii CM 'Ul ^ 72 HACA TM No. 1221^ ON C7\ ••s 9- t .Q -d (D a •p O (D m 8 •H •P Q -d •H H O o •H •PCJ .^ •" O I > CD g •H P P! O ffl ^ tH O o Ch rt -P O •H CJ -d crt •H tI £h O a H Ph P •H tn H 3 H -H OJ (D tn a 0\ er -H ® 03 _ + +^ ocu o P pS © CP P © -H © 03 P! © -H P cd -d fl © fd p to ^ :!^ ■^ O tH OP© O CQ fH tn H CO ON •H P 7i CP © P! •H V © O P © - . CM Si tn ^-^ »t/i p vo © • ,Q ON >5 P Id -d §© o © CO P O • © c3 tiD ;s -d •H cy Cj © © CO G © p m a •H g •H p Ri t>3 O j>- 1o- w CVJ ly^l V\J^ 'Ha 4 ^' N a, fl MCA TM No. 122lf . 73 where r in the argtment is given by r^ = 3C? + y^ + z^ and must not te confused vith the index r which runs through all even numbers . The development (9 -9) is given already hy Morse (reference 6) . One can interpret equation (9'10) as a development of the -Z-functions which contains still an arbitrary parameter rj . For t] — >l there originates, if one divides before by (l - ti^) ' ', the series i^-l)} for r\ — >0 one obtains the series (5 •'+2) for the W-f unctions of the first kind. If one differentiates equation (9 '10) with respect to r\ and sets then ti = 0, there results the series (5»^1) of the V-fvinctlons of the first kind. For C — ^ "" thex-e originates from equation (9 '10) the series (^l-.l) of the X-functions of the first kind. At this point one can recognize why the formulations (i+.l), (i*-.2), (5-1), (5.2), (5-^1), and (5.U2), that is, the series developments considered by Uiven (reference l) all had to lead to the same development coefficients a^. VJhereas equation (9.9) represents a development in terms of eigenfunctions in polar coordinates which have their origin at the point X = y = z = 0, Lame's wave functions can be developed also "in terms of eigenfunctions In polar coordinates ^srlth the origin x=y=0, z=c. This development reads, as shown by a simple calculation, ^l<'U,;rK^^U.;r) - ^ <,.M^r..^U - ^^^^.^C^) (9.n, t=~oo If one miiltiplies by (l - t\^) and then sets ii = 1, equation (9-11) is transformed into the development (5.i)-6). For | — > oo one obtains a development in teims of spherical fxmctlons multiplied by ain (■yr\) and cos iy]) , respectively; the special case of this development m = is already given by Hanson (reference 5) • If one finally develops Lame's wave functions in terms of the eigenfunctions originating by separation of the: wave equation in cylindric coordinates, there results, with the aid of equation (7.21)-), Ih NACA TM No. 1224 ^ cu I H I OJ r ^ H H I •H cd o o CO © o u ID > ,Q O to H csJ a,ri >= (Xi fT" > ■*-o U d s •H ® ^ •d O ra CS (D m o o •H •H O (D ft •H to -P cs5 CO H OJ ktr- P a ti o o •H Cd tsl •H 0) H H ■r^ Rs •H -d -P q +> A:! "H d ta H ■^ IM o 0) © o si m (D H o y o •H ^-v ;i -P OJ liH O (m H go cr\ -p iH -p ^^ o c! _. /3 t>a eg •> © H H O On c3 ■■P © ra •> — od o H -d •^ Pi *> g ^-« © ^-v ra O f^ H H H ra • S • nj g H H ON ra © s o © o S H © • © >' ■p ffl cd H m . . © •§ •H 0:1 ra o •H o <-^ o O Ch H ra cd O -H •H ta -P -H S o © m O -P © © ra «H -p o "§)« •H O § Tb © O -P •P 5h -P Iw ft'H '^ P § r4 Jh O K © H o •H Cd •H -d cd Ph •H 8 •H •P s CP © © o •H -P O m "d © © s cd H m (S © += ^ ^ © d 53 © © Ch © Ch © > ft J^ s2 Ss H m © -d m t> H ?s © •H © -d -P ch © in a ^ B © © P^ "tH M Q 9 © CQ § !>5 ■g > ^ •H 3 & P^ II G? + Ph "^ KACA TM Wo. 122l|. 75 the right side of which is a (recently so--called) Dirac's 5 function. It was introduced first by Soramerfeld (reference 15) and designated by him as prong function. It has a singtilarity at the source point Q, in such a manner that 5(P,Q) dTp = 1 (10.2) /G for each domain G which contains the source point Qj whereas the integral has the value zero if the domain G does not contain the source point. One can interpret 5(P,Q) as limiting case of a function which has for points of influence P in the neighbor- hood of the source point Q a very steep prong whereas it decreases toward the outside very rapidly to zero. The solution of equation (10. 1) is for outside space problems uniquely determined only when besides the boundary conditions on the bounding areas which are at a finite distance an .additional boxmdary condition at infinity is required, namely, the outgoing radiation condition (or else the incoming radiation condition) introduced by Sommerf eld (reference 15) . According to this condition, u(P,Q) for points P at very large distance from the source point Q should behave like an outgoing (or incoming) wave. One designates this solution because of its special properties as Green's function G(P,Q,;k) of the wave equation pertaining to the outgoing (or incoming) radiation condition. Foi* physical reasons the case of the incoming radiation condition will not be considered below. All developments of this section are performed for the coordinates of the prolate ellipsoid of revolution^ one obtains the corresponding formulas for the coordinates of the oblate ellipsoid of revolution by replacing 7 everywhere by iy. Green's function can be developed in the following way in terms of the eigenfimctions of the continuous spectrum G(P,Q;k) = r --i."^— V~u^(lp,T]p,(pp;K)uS(lq^nQ,cpQjK)*<(?')^ (iO-3) i>0 n,m The integration over k goes from to oo, the path of integration deviating at the point k = k in the case of the outgoing radiation 16 IlkCA TM Ifo. 122ij- ■d o § o © ra » © © -P f^ *.— V •ri o ^ d G ?^ •% © ED P! -P -H ^ >s O H f^'H -P o O H -P •H »^— ' G? ed 'd © to a, fJ , *-^ «j+3 n rj a ci] s a Pi P! 2 © © S © %,rt 'H © 3 PI -p O •H i-d -P CQ tI CO cd ^ ^l i < i §§ o ;^ © •P H H P! g ' . _ - J .H ■P Jh Ph -TrQ o !> CU fH •H P< © a o © •H o ■P -P © Xi © o •H f^ c^ m ft, o © u ■p ^ •H © © r! J./1 ,Q ■P SD^ •P 'h o ^^ Q ^ O C,H © © -P •H •H s -d p^ © -H •H Ch P! u © -P 5 CO S -P ca •P O O ^ •N +3 (d ^,,.— ^ t>s O •H © o ■d •H K & ^1— s rt CO ^ -P !h -d -P -p P! rt tiO 9- ?N. •H & § Q •H © a •H o> 1 «•% *• ^ & O g P! cd ^ L, ^ ^ g © R, §3 © J? •H u Cd ■H ^ ^ ■H (0 td ^ !> H © I ^--^ +5 > S © cd © ^ rt O H 3 o O u H :s > ^ ■p •H ^ ~ rt O -P «H-P o JQ -p ,^. > • • •> fv o m «J 3 •H •H © ■p © f^. • •» o & — +=.-^ •rt J^ ^© t= -P © ajn Ph a_n a PS^M 1 © Cd a © •*—»' p- © © O V !d © J^ c3 © iH o •H r-l ^ *■— -V rc; V © O-H Ph ^_^ © O ^ Pi A oo ^— V += ^ fH -P H © CO •H fH '*-—-' H =4 ci "^^ >S Ch §'^ • -d 5i Ph ?i a £h *Ji ,a © O H i _69^ ■ i5i o o m PJ U ^ 'd © c /^ ^**, o © o O d -P ffl' S rPj O o ^ a p! © 'h ri J^ -H •H © M 3 -P -P ^ M a <-— N, o o vH •P !> M l+H ^ O >» O ?^ p! P^ M >s cd cd -H O © ,Q (+^ H -p ^ **-• ••* © H pi ^ Ch Eh o cj 3 1 O' -d -d d © © +3 }>> 3 o fl CO i^ (^ j; © ^ © PJ p; pf o CT^ •H O o > > a © a ^, P! s o 'd •H H cd cd -H © © • , /\ .^ tfl © P( M © P! >5 © © ^ ^ -d o P "^ ^ % f^ A3 P^ © ^ -H h3 >s i> f-i o cd ^g -P oj © ^ ^ '^ © •P O ,Q P) •Hl^ PP © ^ Pi P4 n o o ra ■fi _. KlOJ © O ,C! •P d O o f-J S,'^ J + N Jh 5-; 1> © OJ Q) © tsOH •H © o •P -H •H > • ra © r-^ M © © 11 ^ Is -p a si M d o c ^ 53 •H ^^ ' © ^ -p H O -P ■P f*-^ , -P o +3 m rd f>i a © © ^1 05 u © © ^ .H^ fH © o O & S-d N -P o O t>5 td ^ OD3 Pj. ■gg • a% 1 S3 Ph m s ^ o g S rt O^ Pi ! p3 © tH © PlH O 0) H -H © O CC C CO © ?^ d © MCA TM Wo. 1224 77 u & ••v C5 ra vo ^ ^ • o ra % H ^ CO CD •H +5 S ^ g ^ P" •d 1 ^ e- m a_n •H m 1' T :i3 ■g ^ (D -d i cs" 'f^ (D t^ a-rt • ■\ ;^ P^ P ■*-< , rr t^ O U o ^-"N o g pTH H m s— -■ •H Sj^ «d ^"t. ^— ^ •p K VO -rj ©. ■^^ ^ ^ ra ^•-^ a) H fn ^rf ■ (3) Is cy V o Jk/) p< m (y 5^ ,^-^ XJ\ H (D no 1^ "^^ o r* © > /"\ ^• 5 s f. i 1 •d Xt S c II o ^ '••H Ph -I-' 'H & f Gf m © J^ .P< © © ra PS •H fd 1^ o o • © > ItH © a H •H pi K) •i-l © 5:! ^ o -P © Cm ^ O -d § ^ ei o •p - P4 Pi ^ Jh| ©I Ph ro '-^ Ch i+H 78 NACA TM IIo. 1224 00 CA 09 • • jj o o o H H •H • CQ •H •P O P^ ® ^^ ^.w^ •H 3 fH K !>s -P o eg CO •H Ch rQ O -P ^ to II © "\ -p to •\ Ch u SD >:, -d oa ^ ccj • -d CD. © -H a © •H © CM A -P CO ;h •P 8 CO % + +3 o © ^-, © g ' d •d ^ Ch Pi CM !h H O /■\ II -H 0! ^ O m lA^ "D Ph © P! ^ CJ © ^'■V P" H fl o "— ^ N CO Ph ,Q O •H >> rt J^ •H H ALH p ^ •ri +5 '-^ m (D !h • O t> ?H •P Cd -a- •H Eh C^ ^^ + I—- P< •H -d •H o u o O' Ch p! a H (D ^ — V cd 03 rr *— s 2 o O H ^'""^ ^ ^ g ^ ^ % -p o © +3 rci •S^ 'C3r o •H -P + 1 H Ph g r? -§g -p 9- 03 s cd o a ^ © pi & & f^ -p -d ©^-v -p i ^ D' Q- ^ O cd ^ on Q, CS" 1 © fl © 3 +J • 1*" © % fl © O Ph o o o •H CQ > •H •P ,Q Ch H m ^o^^ o +3 CQ o CO -P O^-' ca •« ?>. o ^3 p! P^ ra >» • •> fd M CVi H O ps S a (D H & © G G? -— ^^ H O Cm o S fl a:) © H O ■P !> cd ■d O' © © -d Pi A 'H •H ^-% ;:S h 1 p- •H M O © O 43 -c! -P H ■P O ?H CQ W) fH 9 a o •H o H 1 © Ch p! Q Pll •rl O "i^ ttl o — — .^ A O s ^ Ph < ^-""'''^ H Pi © O O Ch _. "° m P^ CQ ■d .ja A ,=1 •d ?v •H >r ^■— ^ •H 4^ ffl ^i 54 • •» • CM ^ O O cd •- •H PM -p (3 + Ph CQ CQ 4^ Pi ,d -^ S?^ fj 8- P H Pi iH t*~"^ CQ © •P^-s •H Cy' cd •H Cd ?H fH © ca «.<—>. O s- 1 o H o •H fH O H H Pn-d D2 H jH ^1 Ch c5 Xi H G? § CQ H H Cd A ■P • CVJ ®l V © O O o ~Z^ Cd © u ^ 4^ "^ Cd CQ Cnv_^ ^ Ch ^ CQ Pi p! o ^ '?^ CJ? OJ , O -P fl o • ^ o _, '=' O ••s o p- CS" H O •H !>s-H g§ •H Ph CQ d P! tH -p rd +3 -P OsJI 1 + o o 4^ o o o •H O © CQ 1 •H O cd "" s ■•^Ol (D ,.— V ,s:l © CJ -P B u O fcO fH H +5 -P H Ph o Ch §^■3 S H •H «>..«<' cd —— ;. ALn esJ Ch Ch P H rd g.fn ^c! fl ^ '' ?H •H •H © Ch • i>p O •H Sh ^''^^ Ch tQ 'd 4^ iaD 1 CM fl •rl ■d vC O Ch •- O -H Dp-- •H I -i ^ II II •H Pi © ^ CO © fn CD CQ SO •H o © H Ch ffl ^ fl c 'A o ^"—^ f^,. U cd CfH ta •p q •H a -H ^ • O o •H © •H pj /\. *« © H| O C'J 'd ,£:; ^ += G /\ CJ rt A ■s H Cm 43 u c \J o •H -P s o cd © A A cr K CVI +-"> >5 fd Ch 4^ 43 fcD O (D CM }^ o fiL 1 o O •H C ^ © A Pi :3 •H O il u +5 © O O -P •H & © Pi 5 ^» -' "■* 'd Ol g O rd • -d tD © Ph 1 • O © 'd O C § ^1 © ^d © CQ A ffl ;h .h •\ ^ ■H Vi > ■^ "J^ H G? tl u H ^ © fnH ti fH Pi ©1 o cd A o O •H O +3 t^ +3 03 CO o 5' o © >^ y ,15 H © m PJ f-1 ro ^ Ki •P © > •H H PvH MCA TM No. 1224 79 formed and the contention made that it solves the prohlem of dif- fraction. Actually it represents a \m.Ye which comes as spherical wave from the soixrce point Q, satisfies on the diffracting surface the given houndary condition (as does each single term of the s'om), and which hehaves at infinity like an outgoing spherical wave . If the source point Q, in particular, lies at infinity, Green's functions represent the superposition of a plane wave and of an outgoing spherical wave originating from the diffracting "body mth an amplitude which, in general, is dependent on direction. Treat- ment of the diffraction problem for a source point irithin a finite region is omitted. One starts immediately from equation (IO.8) and contends that the solution of the diffraction problem of a plane wave at the ellipsoid of revolution is given by 2«2^ z ^-W (£,;,,. £«liZl<(3), 5,,,) n m(3) n Ki;r) n X < X^(l)(Tip;7)X^(l)(Tio;y)e^^'''Q"''^^ 1^(7)^ i"^ > i^ b^ J7) (lO.lO) n n n r=-oo in the case of the boundary condition u = for t, = Ci* Under I< (7), one imderstands therein the factor of normalization (9«8) with AA* + BB'^ =1. In the case of the boundary condition 23i = for C = ^1 oJ^® ^^s to replace the two -Z-f unctions with the argument ^1 in equation (10 .10) by their derivatives with respect to 5p at the point ^-j^.' The first term of the svtm in the brackets of equation (10. 10) yields, when the sum over xi,m is formed, exactly the plane wave (10.8); the second term of the sum gives outgoing spherical waves; furthermore, the wave equation and the boundary (surface) condition are satisfied by each separate term of the sum; the contention is therefore proved. For the diffraction at the infinitely thin irire of finite length, one has to set ^1=1. -S^'''''^^ (^t 57) then becomes infinitely large and, in equation (lO.lO), there i-emains only the plane wave. Thus an infinitely thin wire does not present an obstacle for a plane vrave. 80 NACA TM No. 1224 For the diffraction at the infinitely thin circuH.ar disk, y is to te replaced in the formulas TDy iy end t^-, is to "be set eq^^al to zero. ■2 (O^iy) has a finite value so that the outgoing spherical waves do not disappear; that is, even an infinitely thin disk represents an essential disturbance for a plane wave striking it. 11. TABLES 11.1 Comnients to the Tables The tables in section 11.2 contain poorer series developments to y , inclusive ; for the eigenvalue \^{.y) according to n m equation (6.13) and to 7'^, inclusive, for the coefficients a (7) n,r and b (7) according to eqi^ations (6.lJj-) to (6.2l) and equation (U.ll). Furthermore, to 7°, inclusive, the coef- ficients a /a (7), according to eqimtion (7«6), are given for all those cases vhere ^-2l^0) ^-hl^O> ^^^ ^-6/^0 ^issppear- As far as the values of the coefficients ^/^q and b^/b^ are not given in the tables, they disappear; then one must use for the X-f unctions of the second lri.nd tlie series (7'7) aii'i the table for the Oy/s-q' The region of the n- and m-values in the tables extends from m = 0, 1, 2, . . ,,9 and from n =s m, m + 1, . . ., 9« For negative m, vhich are integers, reference is made to the relation (1|-.12) . ■ The last given digit is, in general, probably certain; only where the following digit after rounding up or off, respectively, is a 5, the last given digit votild have to be changed in a fev: cases by unity. In the cases of the end digits ...5, ..50, .5OO, and so forth, it is mostly indicated by a line over or under, respectively, the last digit whether the respective decimal fraction had been originated by rounding up or off. MP.CA TM Ko. 122h 81- For n = the aeries begin to "be useless only at 7 = 10; for larger n thej'- can "be used up to far^larger values of 7 . Below, a few of the first eigenvalues for y'^ = 10 are given as they follow from the exact numerical calculation and from the power series development to 7-^^, inclusive. n 2 k x°(\/io) n ^ ' 2.3050UO 11.790395 25.251313 \° f \/lO j approrimation 2.215 11.880 - -J 25.25ilf7 Figure 1 gives a survey on the dependence of the lowest eigenvalues on 7. The tables in section 11. 3 are taken from the thesis of Bouwkamp (reference 10) . They contain the eigenvalues X°(7) for a number of pairs of values n, 7^, and tlie coefficients a"^ (7) n,r of the pertinent X-f unctions . These latter are fixed so that 2n + 1 2n + 2r + 1 r=-oo n,r (7) = 1 (11.1) The integral of normalization then (compare equation (7.19)) has the value nl .- -■j2 2n + 1 -(1) These tables contain further the values X^"^' (l;7) (11.2) .(1) and X^^'(0;7) (1) I for even and dX^ '(0^7) / d| for odd n. The signs of the a^ are different from those of Bouwkamp since the present series (^.l) and (i!-.2) contain in the coefficients a factor i^ i/hich is missing in reference 10 by Bouwkamp- 82 ]\L\CA TM Wo. 1224 Since the y assume in these tables onlj'' negative values, these functions are appropriate for the treatment of prohlems concerning the ohlate ellipsoid of revolution or for the investi- gation of the eigenvalues of the ion of the hydrogen molecule, vhereas, the tahles in section 11.2 T^ere y can he positive as well as negative, may he uBed for prohlems of the ohlate as well as of the prolate ellipsoid of revolution. Translated hy Mary L. Mahler National Advisory Committee for Aeronautics HACA TM No. 122ij- 83 REFERENCES 1. Niven, C: Philos. Trans. Roy, Soc (London), vol. 171, l88o, pp. 117-151. 2. Strutt, M. J. 0.: Lamesche, Mathieusche imd verwandte Funktionen in Physik und Technik, Ergetnisse der Mathematik und ilirer Grenzge"biete, vol. 1, I932, pp. 199-323. 3. Maclaurin, R. : Trans. Cambridge Philos. Soc, vol. 17, I898, pp. UI-IOS. k. MSglisii, F.: Ann. Physik (k) , vol. 83, 1927, pp. 609-73i^. 5. Hanson, E. T.: Philos. Trans. Roy. Soc (London), A, vol. 212, 1933, PP- 223-283. 6. Morse, P. M. : Proc Nat. Acad. Sciences, vol. 21, 1935, pp. 56-62. 7. Wilson, A. H. : Proc- Roy. Soc (London), A, vol. II8, 1928, pp. 617-635, 635-61J-7. Jaffe, G. : Z. Physik, vol, 87, 1934, pp. 535-51^4. Baher, ¥. G- , and Hasse, H. R. : Proc Cambridge Philos. Soc, vol. 31, 1935, PP- 564-581. Svarthoim, N. : Z. Physik, vol. Ill, I93G, pp. I86-I94. 8. Kotani, M. : Proc Phys.-Math. Soc Japan, III, vol. 15, 1933, PP- 30-57- 9. Chu, L. , and Stratton, J. A.: Joiorn. Math. Physics, vol. 20, 1941, pp. 259-309. See also Stratton, J. A.: Pi-oc Nat. Acad. Sciences, vol. 21, 1935, PP- 51-56, 316-321. 10., Bouwkamp, Ch. J. : Theoretischo en nxaaerieke behandeling van de bulging door en ronde opening. Diss. (Groningen) . Groningen-Batavia 194l. 11. Magnus, W. , and Oberhettinger, F- : Formeln und SStze fiir die speziellen Funktionen der mathematischen Physik. Berlin 1943- 12. Kreuser, P.: tjber das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen. Diss. (Tubingen), Borna-Leipzig 1914. 8k > NACA TM No. 1224 13. Perron, 0.: Die Lehre von den Kettenbruchen, 2. Aufl. , Leipzig und Berlin 1929- lii-. Ince, E. L.: Proc- Edinburgh Boyal Soc, vol- h'J , 1927, pp. 291^.-301. 15. Sommerfeld^ A-: Jahrester- d. Deutschen Math. -Ver., vol. 21^ 1913" pp. ^09-^53- See also Meixner, J.: Math. Zeitschr-, vol. 36, 1933, PP- 677-707. WACA TM Wo. 122if 85 11.2 -Eigenvalues \yy) and Development Coefficients a^^ -^Kt), bj^ -^\l)'> Represented by Brokenr-Off Power Series in 7 TABLE 1. f:^ r) - n(r- + 1) x 10 10 AS POWER SERIES IM 7 m = 0, n = +333^3333337^ -11*811*811*87'^ +1*7031167^ +1358687^ -21*280. 96 r-"-" n = 1 +6000000000 - 685711*29 - 609521+ + 25896 + 872.80 n = 2 +5S 38095238 +101500918 -1*760812 -IU1089 +21*1*12.17 n = 3 +5111111111 + 3291*1763 + 595989 - 2651*2 _ 887.76 n = It +506U93506^ + 17750507 + 5311*7 + 501*0 _ 132.19 n = 5 +50ii27350it3 + 11298966 + 11653 - + 577 + 11*. 80 n = 6 +5030303030 + 787't't3it + 3655 + 150 + 0.91* n = 7 +502262ltlt3li + 581912I+ + 11*13 + 53 + O.li n = 8 +501751^3860 + 1+1+82651 + 628 + 22 + 0.03 n = 9 +5011t005602 + 35621*1+0 + 310 + 11 + 0.01 m = 1, n = 1 +20000000007^ - 1+5711*2867** +121901+87^ - 21031*7^ _ 205.7l7-'-° n = 2 +It28571it286 - 38872692 + 11+1*21*0 + 5682 — 76.28 n = 3 +it66666666i + 1361+7587 -1182501* + 21357 + 229.72 n = \ +1^805191*805 + 119021*17 - 131503 - 5669 + 78.30 n = 5 +It87179it872 + 889I+601+ - 31166 - 337 - 23.61 n = 6 +1+909090909 + 6698782 - 10150 - 26 — 1.93 n = 7 +1+932126697 + 5171*730 - U008 + 6 — O.3I* n = 8 +U9U7368it21 + 1*0991+03 - 1807 + 7 - 0.08 n = 9 +1+957983193 + 3320032 - 899- + 5 - 0.02 m = 2, n = 2 +ll+2857lit297^ - I9i+363't67'' + 3606007^ - 58227^ + 57.7171° n = 3 +3333333333 - 221*1+6689 + 127901 + 607 — 22.60 n = It +1+025971+026 - 2139871+ - 309653 + 6053 — 56.26 n = 5 +1+358971*352 + 2581+89I+ - 101+857 - 532 + 23.27 n.6 +1+51+51+51+51+5 + 31*76672 - 39UOI - 205 — 1.22 n = 7 +Ii660633it8!t + 3361+211 - 16771 - 67 - 0.58 n = 8 +1*73681+2105 + 3005596 - 7911 22 — 0.30 n = 9 +U789915966 + 2620831+ - 1+052 8 - 0.07 m = 3, n = 3 +11111111117^ - 99763067"* + 1326387^ - 16987^ + 17.807!° n = It +2727272727 - 138701*27 + 761+21 - 83 - i*.5l n = 5 +350U273501+ - 1*92001*0 - 92311 + 1750 — 18.22 n = 6 +3939393939 - 877352 - 51*1*68 + 119 + 1*.55 n = 7 +1+20811+1+796 + 755560 - 27781 32 + Oo53 n = 8 +1+385961*912 + 1369050 - 11+1+ 31+ 27 — 0.01 n = 9 +1*509803922 + 151*8920 - 78I+3 11* - 0.01+ m = 1*, n = Ij + 9090909097^ - 577931*57'* + 573167^ - 5787^ + 5.2^7^° n = 5 +2307692308 - 9103323 + 1*1*360 - 118 - 0.79 n = 6 +3090909091 - 1*839057 - 28865 + 570 - 5.53 n = 7 +3571*660633 - 2037903 - 26567 + 129 + 0.73 n = 8 +3891173681*2 - 53051*0 - 17172 + 17 + 0.29 n = 9 +i! 11761*7059 + 21+1*1*18 - 101*67 1* + 0.06 m = 5, n = 5 + 7692307697^ - 36U13297'* + 278837^ - 21*1*7^ + 1.707^° n = 6 +2000000000 - 6271+510 + 261*19 80 — 0.07 n = 7 +2760180995 - 1*157529 - 8311 + 205 - 1.82 n = 8 +326315789^ - 2301599 - 12785 + 79 + 0«03 n = 9 +36131+1+5378 - 10961*95 - 10175 + 22 + 0.10 m = 6, n = 6 + 6666666677^ - 21*1*00877'* + 11*81*07^ - 9778 + 0.627^° n = 7 +1761+705882 - 1*1*9931*1 + 16310 - 1*9 + 0.01+ n = 8 +21+91228070 - 31*1*0673 - 1336 + 80 - 0065 n = 9 +2997198880 - 2221592 - 6051+ + 1*1* - 0„07 m = 7, n = 7 + 5882352967^ - I7ii*035r'* + 81*737^ - 1+67^ + 0.217^° n = 8 +15789't7368 - 33321*31 + 101*35 - 29 + 0.05 n = 9 +2268907563 - 2822595 + 967 + 32 — 0.25 m = 8, n = 8 + 5263157897^ - 121+96627"* + 51177^ 237^ + O.U7!° n = 9 +11+285711*29 - 2535176 + 6898 18 + 0.03 m = 9, n = 9 + 1+761901+767^ - 93895l*>'* + 32367^ - 137^ + 0.027^° 86 mCA TM Noo 122i^ TABLE 2.- n.2 ,(r) a"" (7) X 10-'-° AS POWER SKRIES IN y in.= 0, n = +11111111117^ -352733697** -10190087^ n = 1 + UOOOOOOOO + 3555556 - 151062 n = 2 + 2ltlj897959 + 302901* + 208818 n = 3 + 1763668113 + 66996 . + 1*6389 n = 4 + 1377l*10lt7 + 21683 + 18710 n = 5 + 112963959 + 8738 + 91*67 n = 6 + 9572611.96 + 1*071 + 51*62 n = 7 + 8301+1*983 + 2105 + 31*39 n = 8 + 73325729 + 1177 + 2306 n = 9 + 656U0291 + 700 + 1622 m = 1, n = 1 + 1333333337^ - 35555567 ** + 6131*87^ n = 2 + 1221*1*8980 - U5U356 - 17898 n = 3 + 105820106 - 120593 + 16210 n = U + 91827365 - 1*3366 + 9870 n = 5 + 8068851*2 - 18723 + 5930 n = 6 + 71791+872 9160 ■1- 3767 n = 7 + 61*59051*2 1*912 + 2521+ n = 8 + 58660583 2826 + 1768 n = 9 + 53705693 1719 + 1285 m = 2, n = 2 + 1*08163277^ - 7572607** + 122277^ n = 3 + 52910053 - 3011*82 - 11*30 n = 4 + 550961*19 - 130098 + 2386 n = 5 + 53792361 - 621*11 + 251*1 n = 6 + 51282051 - 32716 + 2057 n = 7 + 1*31*1*2907 - 181*20 + 1579 n = 8 + 1*5621*898 - 10990 + 1207 n = 9 + 1*2961*551* - 6877 + 931 m = 3, n = 3 + 17636681*7^ - 231*1*867** + 30037^ n = U + 2751*8209 - 151781 + 165 n = 5 + 32275IH7 - 87376 + 621 n = 6 + 31+188031* - 50891 H- 828 n = 7 + 31*602076 - 30700 + 812 n = 8 + 31*218671* - 19232 + 718 n = 9 + 331*16876 - 121*81 + 609 m = U, n = I4. + 91827367^ - 910687** + 9187^ n = 5 + 16137708 - 78638 + 210 n = 6 + 20512821 - 51*963 + 21+2 n = 7 + 23O68O5I - 3681*0 + 329 n = 8 + 21*1*1*1910 - 21*727 + 361 n = 9 + 25062657 - 1681*9 + 353 m = 5, n = 5 + 53792367^ - 1*11927** + 3327^ n = 6 + 102561*10 - 1*3181 + 131 n = T + 1381*0830 - 31*735 + 118 n = 8 + 16291*606 - 25905 + 152 n = 9 + 17901898 - 18912 + 178 m = 6, n = 6 + 3'»188037^ - 207937** + 1367^ n = 7 + 69201*15 - 25086 75 n = 8 + 9776761* 221*51 + 61* n = 9 + 11931*598 - 18212 + 79 m = 7, n = 7 + 2^0680=7^ - 111+037** ■ 627^ n = 8 + 1*888382 - 15307 + 1'3 n = 9 + 7160759 - 11*900 + 37 m = 8, n = 8 + 16291*617^ • - 66727** + 307^ n = 9 + 3580380 9-7U3 + 25 m = 9, n = 9 + 11931*607^ - 1*111*7** 16^^ NACA TM No. 1224 87 TABI£ 3.- — i X lO-"-" AHD n,o X 10 10 AS fOfSR smxss ai^/^o XIOIO a6/«o X 10^° m = 0, n > +I90U76197'* -7.696017^ +1371*297^ n = 1 + '*5351'^7 + 55817 + 21*667 n - 2 + 206U3I + 371*0 + 3970 n = 3 + 117727'* + 681^ + i*26g n = h + 760700 + 189 + 2355 n =. 5 + 53159^ + 67 + 11*37 n - 6 + 3922$0 + 28 + 91*0 n =. 7 + 30121^2 + 238563 + 13 + 61*9 n :. 8 + 7 + 1*66 n :. 9 + 193571 + U + 31*6 a = 1, n = 1 + 9070297"* - 331*907^ + 3521*7^ n = 2 + 68711*1* - 371*0 + 221*2 n « 3 + 501*51*6 - 879 + 11*21 n . k + 38O35O 281* + 9U2 n > 5 + 295331 Ul + 653 n . 6 + 235350 52 + 1*70 n - 7 + 191701 2U + 31*9 a > 8 + 15901*2 13 + 266 n = 9 + 131*010 7 + 208 « = 2, n :i= 2 + 1371*297'* - 371*07^ + 3207^ n . 3 + 168182 - 11*66 + 355 a ^ k + 163001 608 + 311* n = 5 + 11*7662 - 277 + 261 n = 6 + 130750 - 138 + 211* n . 7 + 115021 73 + ITS n = 8 + 101209 1*2 + 11*3 n > 9 + 8931*0 21 + 119 n = 3, n = 3 + 336367** - 681*7^ + 517^ n ~ It + 51*336 - 1*73 + 79 n - 5 + 6328s 277 + 87 n > 6 + 65375 - 161 + 85 n = 7 + 63900 95 + 79 n = 8 + 60725 58 + 72 n . 9 + 56853 37 + 61* m = U, n = U + ioS6xr^ - 1707^ + 117^ n = 5 + 21095 166 + 22 n = 6 + 2801S 121* + 28 ■ n = 7 + 31950 86 + 32 n = 8 + 33736 58 + 33 n = 9 + 3lvll2 I4.0 + 32 m = 5, n = 5 + 1*2197"* - P' + 37^ n ^ 6 + 9339 61 + 7 n = 7 + 13693 58 + 11 n = 8 + 16868 1*6 + 13 n = 9 + I8951 21 + IS ji = 6, n = 6 + 18687'* 197^ . 176 n = 7 + U561* 28 + 3 n = 8 + 722a 28 + 1* n = 9 + 91*75 21 + 6 B = 7, n . 7 * 9137** - .876 * 076 n = 8 + 2I1.IO 13 + 1 n = 9 + UO61 11 + 2 m = 8, n = 8 + 1*827'* - ■ 376 + 076 n = 9 + 1351* 6 + m = 9, n = 9 + 2717"* 276 . 076 88 HACA TM No. 1224 TABI2 1+.- °* a (7) X 10 ,10 AS PCHSR SERIE IB 7 m = 0, n = 2 - 2222???22r + 7051*671*7'^ + 1519717^ n = 3 - 1711*28571 - 1523810 + U2927 n = 1+ - 13605hh22 168280 - 26l5fi n = 5 - 11223 3'*'i6 1*263U 821*1 n - 6 - 95359186 15011 1*1*52 n = 7 - 8281*0237 61*08 2790 n = 8 - 7320261U 3113 I889 n = 9 - 65561828 1662 131*6 m = 1, n = 1 -33333333337^ -13333333337'* -51*09523817^ n = 2 - 666666667 - 631*92063 - 61*78782 n = 3 - 31*285711*3 + 911*2857 - 197018 n = k - 226757370 + 81*11*00 + 16156 n ^ 5 - 168350168 + 191852 7170 n . 6 - 133502861 ♦ 6301*7 5138 n = 7 - 1101*5361*9 + 25630 33«1 n = 8 - 91*11761*7 + 12009 2295 n = 9 - 81952286 + 6232 1621 m = 2, n = 2 -13333333337^ - 631*9206357'' - 805528907^ n = 3 - 5711*28571 + 761901*76 + 8760622 n = h - 31*0136051* + 6310502 122277 n = 5 - 235690236 + 131*2961* 1*268 n = 6 - 178003811* + 1*20316 1*295 n = 7 - 11*2011831* + 161*766 3369 n = 8 - 11761*7059 + 75051* 21*1*9 n = 9 - 10016390^ + 38087 1778 m = 3, n = 3 - 85711*28577^ + 2666666677'' + 883800917^ n = l^ - 1*761901*76 + 20611*306 - 21*51288 n = 5 - 311*25361*8 + 1*178U1 61*139 n = 6 - 2288620117 + 12609I4.8 7360 n - 7 - 177511*793 + 1*80568 3572 n = 8 - 11*37908^ + 211*01*3 21*75 n = 9 - 120196685 + 10661*2 1829 m = U-, n = It^ - 631*9206317^ + 1*91*71*3357'* - 1295161*87^ n = 5 _ li.0U0l*0l*0l* + 966931*3 - 380101* n = 6 - 286077559 + 2837133 31*1*29 n = 7 - 2169625^ + 105721*9 685i n = 8 - 17251*9020 + 1*62332 2950 n = 9 - 1U2050628 + 226858 1908 " = 5, n . 5 - 5050505057^ + 189933527'' - 13658317^ n . 6 - 31*9650350 + 51*1*9096 - 123721* n - 7 - 260355030 + 1993669 191*30 n > 8 - 203921569 + 858612 5180 n . 9 - 165725733 + 1*15905 231*7 » = 6, n = 6 - 1*195801*207^ + 91*1*51007'* - 31*70117^ n - 7 - 307692308 + 31*03331* 521*56 n . 8 - 2379081*97 + 11*1*6929 11529 n = 9 - 191222000 + 693176 3762 m = 7, n = 7 - 358971*3597^ + 51*1^3957"* + 2276636 - 1238787^ n = 8 - 271*509801* 25809 n - 9 - 2185391*28 + IO8027I* 7151* jB =r 8, n = 8 - 3137251*907^ + 31*021*1*67'* 537207^ n = 9 - 21*7678019 + 1601021 11*025 m = 9, n = 9 - 2786377717^ ♦ 22811*567'* - 265037^ NACA TM Wo. 122ij- 89 a 1,(7) ,- TABia 5.- -^'^ X 10-^° AHD s(r) <,o^y^ ^t^ — X 10^° AS powm ssRiss iH r a' ii) ^-kK xlol° a^/a^ X 10^ m = 0, n = l^ + 9070297'^ 67317^ n = 5 + 68711^1^ + 2610 n = 6 + 501*5^ + 371 16027^ n = 7 + 380350 + 106 1207 n = 8 + 295331 + 39 852 n = 9 + 235350 + 17 610 m = 1, n = 3 + 190l^76l9r'* + 338621+7^ n = U + l^5351'*7 + 100968 - U58107^ n = 5 + 2061U3I 231+92 n = 6 + 117727'i 2592 - 11212 n = 7 + 760702 635 - 1+830 n = 8 + 531592 213 2557 n = 9 + 392250 86 1521+ m = 2, n = 2 -666666667>'* -126931+1277^ n = 3 + 95238095 + 8I+65608 + 15117167^ n = U + 13605^^1+2 + 1511+520 n = 5 + U8IOOO5 - 271+071+ - 320667 n > 6 + 2351151+5 - 259V5 - 1+1+81+9 n = 7 + 1369260 5719 - 11+1+90 n = 8 + 885992 1776 6392 n = 9 + 616393 677 3353 m = 3, n = 3 +28571'+2867'^ + 592592597^ -3171+60327^ n = 1* + 317'^6032 + 821+5723 +10582011 n = 5 + 9620010 - 1279011+ - 1231+668 n = 6 + 1+238186 - 106971 - 131+51+6 n = 7 + 2282100 2221+1 - 36221+ n - 8 + 1392273 6511+ - 11+061 n = 9 + 921*590 2370 6705 m =• U, n = li + 63I+920637 + 29681+6017^ +1+232801+27^ n' = 5 + 17316017 - 1+11+1+001+ - 38I+8OOI+ n = 6 + 70636I4.3 - 326912 - 336361+ n - 7 + 3586157 62911 - 79692 n . 8 + 20881+09 17587 - 28127 n . 9 + 1335519 6162 - 12U52 » - 5, n - 5 + 288600297"* - 108533'^l+7^ - 96200107^ n = 6 + 111 00011 - 80727I+ - 7*0001 n - 7 + 5379236 - 11+8289 - 159385 - 52235 n . 8 + 3016591 39912. n = 9 + 1869726 13555 - 21792 m = 6, n = 6 + 166500177"* - ^90937^ - 11+800017^ n = 7 + 7770008 - 309391* - 296000 n - 8 + 1+223228 80721 - 911+12 n = 9 + 251+9627 26700 - 36319 a = 7, n = 7 + 108780117"* - 5906617^ - 5180017" n = 8 + 575891+7 - 150108 - 152353 n = 9 + 3399502 1+851+6 - 58111 m = 8, n = 8 + 77102387"' - 2628057^ - 21*1*7697^ n = 9 + Ui+i+5503 83016 - 89808 m = 9, n = 9 + 57156I+77"* - 1351977^ - 131+7127^ 90 NACA TM No. 122ij- TABLE 6.- NUMKRICAL VALUES OF THE COEFFICIENTS m / m APPEARING IN THE SERE^ DEVELOPMENTS (T-T) "■-2/^0 ^ ^'^^^ m = 0, a = n = 1 +50000000007^ -6666666677^ +661375667^ -1666666667 +222222222 -16825397 ^--^/^o ^ 10-^° m = 0, n = +1666666677 n = 1 -277777778 n = 2 + 55555556 n = 3 + 14-761905 m 1, n = 1 +2777777787 n = 2 -166666667 -21+6913587 +3171+60 32 - 705^^67 - 28219 +952380957^ - 6 31+9206 °--6S X loio m = 0, n = +17636687^ n = 1 -5291005 n = 2 +3703701+ n = 3 - 529101 n = 1+ - 50391 n = 5 - 7632 m = 1, n = 1 +265957^+7^ n = 2 -3703701+ n = 3 +1058201 n = 1+ + 251953 m = 2, n = 2 +71+07^077^ n = 3 -5291005 WACA TM Wo. 1224 91 TABLE 7.- -^ D"" (r) n.o X 10-^" AS POWER SERIES IB 7 [Tor m = 0, there la \ = b^; compare therefore table 2 J m = 1, n - 1 +8000000007^ -213333337^ +3680897^ n = 2 +14.08163265 - 151'*520 - 59660 n = 3 +261*55026^ - 3011*82 + 1*0526 n = 1* +192837*^66 - 91068 + 20727 n = 5 +150618612 - 3'^950 + 11069 n = 6 +123076923 - 15701* + 6U58 n = 7 +103806228 7891* + 1*057 n = 8 + 89620336 1*357 + 2702 n = 9 + 787683U9 2522 + 1882 m = 2, n = 2 +61221^1^8987^ -113589037'' +1831*01*7^ n = 3 +370370370 - 2110372 - 10012 n = h. +257116621 - 607123 + 11136 n = 5 +193652501 - 221*681 + 911*9 n = 6 +15381^615^ - 9811*7 + 6171 n = 7 +12687't279 - 1*821*3 + 1*135 n = 8 +1075i(-i*^i4.03 - 25902 + 281*1* n = 9 + 93089868 - 11*900 + 2017 m = 3, n = 3 +1*938271607^ - 656560 37*" + 81*0717^ n = li +330578512 - 1821369 + 1978 n = 5 +21*2065627 - 655319 + 1*651* n = 6 +1880 314 188 - 279902 + 1*553 n = 7 +15221*9131 - 135080 + 3571* n = 8 +127097931 - 711*32 + 2666 n = 9 +108601*81*5 - 1*0562 + 1980 m = l+, n = It- +1*1322311*07^ - 1*0980817** + 1*12937^ n = 5 +295857988 - 11*1*1703 + 3851 n = 6 +22561*1026 - 601*588 + 2666 n = 7 +179930796 - 287353 + 2567 n = 8 +11*8280919 - 150013 + 2193 n = 9 +125313283 - 81*21*1* + 1766 m = 5, n = 5 +3550295867^ - 27186397*^ + 218807^ n = 6 +266666667 - 1122807 + 31*10 n = 7 +209919262 - 526811* + 1790 n = 8 +171093368 - 272002 + 1601 n = 9 +11+3215181 - 151296 + 11*22 m = 6, n = 6 +3111111117^ - 18921387 '^ + 123857^ n = 7 +21*2211*533 - 878023 + 262^ n = 8 +195535278 - 1+1*9019 + 1285 n = 9 +162310538 - 21*7677 + 1072 jn = 7, n = 7 +2786166097^ - 136831*77'' + 71*107^ n = 8 +22160661*8 - 693939 + 1939 n = 9 +182599356 - 379959 + 952 m = 8, n = 8 +21*93071^797^ - 10208917"* + 1*61*37^ n = 9 +201*081633 - 555321* + 11*22 a » 9, n = 9 +2267573707^ - 7815677'' + 30257^ 92 mCA TM Wo. 1224 TABLE 8.- '^«'* ^- X 10^° AHD "t^ ^ - x 10^° AS POWIR SBKIES IH 7 b? J7) b" (r) |ror m = 0, there Is bj. = ar; coMpere therefore table 3 3] hjh„ X lOlO hg/b,, X 10^0 m » 1, n = 1 +136051*1*27'* -5023517 +986677^ n = 2 + i(.8l000i - 26177 +26909 n . 3 + 2351*51*3 - UlOl* +10651* n = U + 1369260 - 1C21 + 5181 n » 5 + 885992 - 333 + 287U n = 6 + 616393 - 130 + 171*6 n - 7 + 1*51867 - 58 + 1135 n = 8 + 3Uii592 28 + 777 n = 9 + 270999 - ^1 + 551* m » 2, n = 2 + 96200107'' -2617697^ +672737^ n - 3 + 1*238186 - 36931* +231*39 n = U + 2282100 - 8508 +10362 n = 5 + 1392273 - 2615 + 5337 n - 6 ♦ 921*590 - 971 + 3056 n - 7 + 652697 - 1*17 + 1892 n . 8 + U821*28 - 198 + 121*3 n - 9 + 3695iilt - 102 + 856 m - 3, n . 3 + 706361*37'' -11*36337^ +u6e787^ n . U + 3586157 - 31198 +192U- n =. 5 + 208&*09 - 9151* + 9339 n = 6 + 1335519 - 3281 + 5091* n = ^ + 913776 - 1362 + j027 n = 8 + 657857 - 630 + 1921 n » 9 + 1*92725 - 317 + 1281* B = It, n = U + 53792367'' - 81*2337^ + 336787^ n . 5 + 3016591 - 23800 + 15565 n = 6 + 1869726 - 8277 + 8150 n = 7 + 121*6058 - 331*3 + ii67g n = 8 + 87711*2 - 1512 + 2881 n = 9 + 61*1*333 — 71*7 + 1876 m =■ 5, n = 5 + U2232287'' - 523597^ +2U90^7^ n = 6 •t- 251*9627 - 17737 +1259S n = 7 + l66luii - 700^ + 7017 n = 8 + IIU7032 - 310-' + 1*211 n = 9 + 8281*28 - 1509 + 2680 m = 6, n » 6 + 33995027'' - 31*1597^ +188927^ n = 7 + 2172611* - 13231 +10256 n = 8 + 1^71*756 - 5770 + 6016 n = 9 + IOU93U2 - 2761 + 3752 m >■ 7, n = 7 + 2-^33617'' - 211977 +1^6517^ n = 8 + 186802U - 9966 + 8U23 n = 9 + 13116-^8 - 14706 + 5160 m = 8, n = 8 + 2 3 350 307'' - 162907^ + 115817^ n = 9 + 1620308 - 7602 + 6981 m = 9, n = 9 + 19803767" - 117697^ + 93087^ MCA TM No. 1221+ 93 TABLE 9. V X 10^0 AS POWER SERIES IK y [For m = 0, there is bj, = a^.; compare therefore tahle l4_j m = 1, n = 3 -571^28577^ +15238107^ -328367^ n = U -68027211 + 252U20 + 1+81+7 n = 5 -673^0067 + 7671^1 - 2868 n = 6 -63572791 + 30022 - 21+1+7 n = 7 -59171598 + 13730 - 1811 n = 8 -5i<-90196l + 7005 - 1339 n = 9 -509925 3^^ + 3878 - 1009 m = 2, n = i+ -226757377^ + 1+207007^ - 81527^ n = 5 -336700 3i+ + 191852 - 610 n = 6 -381U367U + 90068 - 920 n = 7 _39Ui+7732 + 1+5768 - 936 n = 8 -39215686 + 25018 - 816 n = 9 _382Ui4-i4-00 + 1U5U2 - 679 m = 3, n = 5 -112233*^^7^ + 11+92187^ - 22917^ n = 6 -19071837 + 105079 - 655 n = 7 -23668639 + 61+076 - 1+76 n = 8 -26114-3791 + 38917 - 1+5U n = 9 -273171+28 + 21+237 - 1+16 m = 1+, n = 6 - 63572797^ + 6301+77^ - 7657^ n = 7 -11831+320 + 57668 - 371+ n = 8 -1568627^ + 1+2030 - 268 n = 9 -18211619 + 29081+ - 21+^ m = 5, n = 7 - 391^1^7737^ + 302077*^ " 291+7^ n = 8 - 781+3137 + 33021+ - 199 n = 9 -10926971 + 271+22 - 15^ m = 6, n = 8 - 2611+3797^ + 159007^ - 1277^ n = 9 - 51^631+86 + 19805 - 107 m = 7, n = .9 - 18211627^ + 90027^ - 607^ For n = m and n = m + 1, h_2/bQ disappears. 9^^ mCA TM No. I22U m m TABLE 10.- ^L^XIOIO AHD ^^^6fll^iolO AS n,o t" (7) ja. ir) PCWER SERIES IN 7 [For m = 0, there is 'bj, = a^; compare therefore table 5J U^o X lolO ^-6K X 10^° m = 1, n = 5 n = 6 n = 7 n = 8 n = 9 +1371+297^ +168182 +163007 +IU7665 +130750 -15667^ - 371 -136 - 59 - 29 -17276 -213 -203 m = 2, n = 6 n = 7 n = 8 n = 9 + 336367^ + 51+336 + 63285 + 65375 -3717^ -227 -127 - 72 -307^ - 51 m = 3, n = 7 n -■ 8 n = 9 + 108677^ + 21095 + 28018 -1067^ - 99 - 72 - 776 m = l+, n = 8 n = 9 + U2197^ + 9339 -3676 -1+3 m = 5, n = 9 + 18687^ - ikr^ For n = m, n = m+l, n = m+2, n = m+ 3, h.j^/bQ disappears. For n=m, n = m+l, n = m+2, n=m+3^ n=m+i+, n = m + 5, "b^g/bQ disappears. KACA TM No. 122lj- 95 11.3 — Numerical Magnitude of the Eigenvalues and the DeTelopment Coefficients for Different n,7 and m = TABLE 11 .- EIGENVALUES X°(r) n -r^ '^o Xi X2 ^3 3 - LUl^SS^^ + O.lit-0119 + l^. 530790 +10 .U9U513 1^ - 1.59^507 - .5052i^ij- + u. 091201 +10.00386U 5 - 2.079939 - I.162U22 + 3.677958 + 9 .517981 6 - 2.599717 - 1.831051 + 3.288927 + 9.036338 7 - 3.151917 - 2.510^^21 + 2.92331U + 8.558395 8 - 3.731+090 - 3.200050 + 2.578205 + 8.083615 9 - i+. 3^+3^+39 - 3.8991^00 + 2.250701+ + 7.6111+65 10 - u. 976896 - U. 607952 + 1.938379 + 7.11+11+28 15 - 8.U2084 - 8.2718O + .U99I+9 + 1+ .80616 20 -12.1629U -12 .0991^3 - .91I27 + 2.1+5867 25 -16 .0790i+ -16 .0501^1 - 2 .1+1^60 + .06093 50 -36 .90015 -36 .89912 -13 .5651+8 -13 .21675 100 -81 .0279^ -81 .02791+ -45 .1+8967 -1+5 .1+8391 2 -7 Xi, X5 ^^6 ^7 U +28.00092 5 +17.511596 +39 .501+1+99 10 +15.1131+2 +25 .0691+9 +37.01+822 15 +12 .81726 +22 .68771 +31+. 63123 20 +10 .61+631+ +20.36028 +32.25386 25 + 8.6301+0 +18.081+57 +29 .91689 50 + .91+568 + 7.25075 +18.92267 100 -16 .06556 -15.32812 + 2.57368 +11 .1+5561+ ^2 ^^8 X9 ^10 ^12 25 100 +59 .736180 +26 .561+08 +1+3 .1+9371+ +97.652659 +62 .82728 +11+3 .606898 96 MACA TM No. 122i4- TABLE 12.- HUMEFICAL VALUES OF THE COEFFICIEKTS a° (7), AM) o.r THE FUNCTION VALUES X^-^'(l;7) AND X^-^^(0;7) r -72 = 3 k 5 6 2 k 6 8 10 Xo(l) Xo(0) +0.987210 - ,356220 + 18683 ko& + 5 +0.976788 - .1^78331^ + 33565 979 + 16 +0.963507 - .597277 + 52U82 19 lU + 38 +0 .9i+78i^0 - .7105^5 + 7^937 3279 + 79 1 +1.362526 + .815979 +1.^+89682 + .7i)-9906 +1.615218 + .683961 +1.736681 + .619666 r -r^-1 8 9 10 2 k 6 8 10 +0.930i;29 - .816037 + .100273 511^ + li^i^ 3 +0 .911986 - .912632 + .127817 -jhkO + 238 5 +0.892960 - .999698 + .156891+ - ■ 10250 + 369 8 +0.871+035 -1 .0771+18 + .18691+3 - 13535 + 51+1 11+ Xo(l) +1.852000 + .5581+52 +1 .960118 + .501339 +2.060379 + .kkeskG +2.1521+86 + .1+0131^5 r -72 = 15 20 25 50 100 H^. 78915 +0.72576 +0 .67909 +0.55601 +O.U6036 2 -1.31+978 -I.U9587 -1.57800 -1 .68750 -1 .61+671+ k + .33881 + .1+7816 + .60025 +1.01U70 +1.38U12 6 - 3608 - 6622 - .10110 - .29658 - .63255 8 + 260 + 515 + 965 + 5161 + .181+83 10 10 26 59 - 595 - 371+5 12 + + 1 + 3 + 1+8 + 555 11^ 3 63 16 + 1 Xo(l) +2.51651+ +2.7711+2 +2 .96871 +3.61286 +1+. 35229 Xo(0) + .23073 + .13779 + .08609 + .01281+ + .00081 MCA TM No. 122ij- 97 TABLE 13.^ NUMERICAL VALUES OF THE COEFFICIENTS a° (7), AND THE FUNCTION VALUES X^-^^(l;7) AND dLX^-^-^Oir) /il r -^^ = 3 1+ 5 6 2 1^ 6 8 +0.997105 - .116098 + 3902 63 + 1 +0 .9914-981*- - .152711 + 6812 150 + 2 +0.992380 - .18801^8 + 9751 262 + 1+ +0.989330 - .222236 + 11+717 1+73 + 9 Xi(l) Xi(0)« +1.117169 + .830138 +1.151^659 + .778367 +1.1901;U5 + .728028 +1 .226765 + .682557 r -r^-1 8 9 10 2 1+ 6 8 10 +0.985910 - .255039 + 19595 753 + 16 +0 .982170 - .2861+70 + 25008 1066 + 27 +0.978150 - .316613 + 30907 11+77 + 1+2 1 +0.973908 - .31+5385 + 37211 1969 + 62 1 Xi(l) Xi(0)« +1.261293 + .638528 +1.291+71+1 + .597088 +1.327190 + .558051 +1.358536 + .5211+1+1+ r -7^ = 15 20 25 50 100 +0.95067 +0.92651+ +0.90339 +0.81333 +0.71269 2 - .1+7019 - .56735 - .61+300 - .8i+i+i+3 - .95736 1+ + 7352 + .111+08 + .15571 + .31+108 + .57527 6 572 - 1160 - 1935 - 7563 - .20752 8 + 27 + 71 + 11+6 + 1060 + 5010 10 1 3 T 102 - 863 12 + 7 + 111 11+ 11 16 + 1 Xi(l) +1.50038 +1.62031 +1.72298 +2.08616 +2.51279 Xi(0)« + .37136 + .26571 + .19192 + .01+1+30 + .001+21 98 MCA TM Wo. 122^1- TABLE Ik.- NUMERICAL VALUES OF TEE COEFFICIEKTS a° {7), AND THE FUNCTION TALUES X^-'-\l;7) AND J>^\q;7) r -r2 = 3 h 5 6 -2 2 1^ 6 8 +0.071289 + .985721 - 72766 + 18MD 2U + 1 +0 .095777 + .97^1'+8 - 961^30 + 3258 57 + 1 +0.119671 + .959391 - .11965^ + 55067 m + 1 +0.11+21+71^ + .91+1921+ - .11+2393 + 7260 190 + 3 X2(l) XgCo) +0.989062 - .5371^31 +0 .978117 - .51+7702 +0.96U533 - .556050 +0.91+9296 - .562257 r -7^ = 7 8 9 10 -2 2 1+ 6 8 +0.163772 + .922381+ - .161+669 + 9837 302 + 6 +0.183325 + .9011+21+ - .186536 + 12798 1+50 + 10 +O.20103I+ + .87961+2 - .208085 + 1611+8 61+0 + 16 +0.216893 + .8575^+9 - .2291+32 + 19901 880 + 21+ Xgd) 12(0) +0. 9331+26 - .566205 +0.917893 - .567935 +0.90 31+97 - .567698 +0.890893 - .565615 r ^2 = 15 20 25 50 100 -2 +0.27303 +0.30282 +0.31712 +0.28828 +0.22335 + .75076 + .65357 + .55985 + .131+10 - .20363 2 - .33636 - .1+1+827 - .56362 - .91568 - .79121+ 1+ + i+532 + 8373 + .13671 + .50022 + .89250 6 306 768 - 1596 - .12202 - .1+191+9 8 + 13 + J+3 + 112 + 171+5 + .11556 10 1 2 5 166 - 2129 12 + 11 + 21+0 11+ 1 21 16 + 2 X2(l) +0.86260 +0.89086 +0.96019 +1.1+0297 +I.8I57I+ X2(0) - .53563 - .1+8567 - .1+21+31 - .I388I+ - .01312 NACA TM Wo. 1221+ 99 TABLE 15 .- MJMERIGAL VALUES OF TEE COEFFICIENTS a° { 7 ) , AND TEE FUNCTION VALUES X^-'"^(l;7) AMD d3^ ^\0',y) l±i -2 2 1+ 6 8 Xgd) Xo(0)' +0 .0U9768 + .996217 - 52769 + 1057 12 +1.000287 +0 .06'^k'Jk + .99338I+ - 70252 + 1877 27 +1.000066 -1.14-27867 +0.080671 + .989910 - 87661 + 2929 53 + 1 +0.999883 +l.i+07i)-50 +0.095329 + .985786 - .101+980 + 1+212 92 + 1 +O.9997I+2 -1.386162 ^^ -2 2 1+ 6 8 7 +0 .1091+32 + .981091 - .122202 + 5725 11+5 + 2 8 +0.122969 + .975879 - .139316 + 7^+66 217 + 1+ +0.135938 + .970201 - .156318 + 9^33 308 + 7 10 +0.li+83it-3 + .961+106 - .173200 + 11626 1+22 + 10 X3{1) Xo{0)' +0.999733 -1.361+111 +0.999913 -1.314-1380 +1.000329 -1.31801+0 +1.001021 -1.29 1+172 r N. 15 20 25 50 100 -2 +0.20229 +0.21+1+38 +0.27699 +0.35635 +0.371+61+ + .92878 + .88803 + .8I+U16 + .611+60 + .27353 2 - .25562 - .331+23 - .1+0827 - .681+95 - .82807 1+ + 2590 + 1+51+8 + 6992 + .23918 + .56017 6 ll+l 332 61+1 - 1+395 - .19701+ 8 + 5 + 15 + 37 + 508 + 1^388 10 2 1+1 681 12 + 2 + 78 11+ 7 13(1) +I.OO9I+7 +1.02683 +1.05216 +I.23I8I+ +1.53571 x.(o)« -I.169I+9 -1.01+11+7 - .9I58I+ - .1+2163 - .07382 100 KA-CA TM No. 122if TABLE 16.- NtJMESICAl TALOEB Of THE COKFriCIKHTS a° (7), MKD TSS lOTCTIOH VAIilES I, "'• (l;?) n = U r \v 5 10 15 20 25 -it- -2 2 h 6 8 10 + 231^9 + 67563 +0 .99l^2l8 - 68652 + 1897 29 + + 9906 +O.1370U5 + .97591^2 - .136070 + 73^ 231^ + 5 + 22lt0 +0.19952 + .9U636 - .20137 + l68ii T9 + 2 + koKe, +0.26279 + .90051 - .26269 + 2956 - 185 + 8 + 6318 +0.32153 + .Sit069 - .31828 + 1^535 357 + 18 1 TkU) 1^(0) +0.999582 + .388021 +0.992656 + .W3891 +0.98827 + .U1853 +0.97333 + .1^351*9 +0 .91*971 + .1*5131 n = k \. 2 N? \. 50 100 r ^\^ -k- +0.1T281* +0.19^17 -2 + .U5662 + .2511*7 + .hOSkJ, - .27937 2 - .50209 - .60518 1* + .16360 + .5991*1 6 V + 2753 - .2361*0 8 + 291 + 51*58 10 21 8U9 12 + 1 96 11* 8 16 + 1 Xl^(l) +0.82102 +1.1691*1* 14(0 ) + .1*3598 + .11*21*5 n = 5 \^ 10 15 r ^v -1* -2 2 1* 6 S + 651*3 +0.11091*9 + .985522 - .112180 + 5292 11*3 + 3 + ll*2U +0.161*1*1 + .96761 - .16690 + 1185 1*8 + 1 10 15(1) +0 .998731* +0.99668 15(0)' +1.788369 +1.73792 n = 5 V 2 \^ 20 25 50 100 r \,^ -K + 21*31* + 3632 +0.10628 +0.19798 -2 +0 .2151*9 +0 .2631*2 + .1*3712 + .1*9281 + .91*291 + .91191 + .68991* + .18816 2 - .22006 - .27117 - .1*8639 - .68181 1* + 2093 + 321*5 + .I22U3 + .39752 6 111* 221 - 1710 - .11798 8 + 1* + 10 + 160 + 2220 10 10 293 12 + 29 11* 2 xjd) +0.99393 +0 .99071* +0.98672 +1.11609 XjCo)' +1.68269 +1.62227 +1.25100 + .511*19 WACA TM No. 122ij- 101 TABLB 17.- BUMERICAL VAUJKS OF THE COKFFICIHnS b^ ^.h) AS WELL AS OF mE I-^TJBCTIOHS OF TEE FIHST KIHD FOB THE AEGUMEHTS 1 AHB 0, wns n = 6, 7, 8, 9, 10, 12 n = 6 \^2 r \^ 5 10 15 20 25 -6 -1* -2 2 1* 6 8 + 20 + 1257 + 1*7578 +0 .9971*85 - 1*7797 + 980 12 + 1 + 165 + 5001 + 91*691* +0 .989966 - 95268 + 3910 91* + 3 + 57 + 1123 +0 .11*139 + .9801*2 - .11*252 + 879 32 + 1 + 136 + 1979 +0.1860^ ■,+ .96011 - .18796 + 1552 75 + 2 + 268 + 3071* +0 .22952 + .93789 - .23256 + 21*10 - 11.5 6 X6(l) 16(0) +0.999935 - .317371* +0.999383 - .3221*30 +1.00132 - .32872 +0.99675 - .33331 +0 .991*60 - .33921* n = 6 \ 2 N?" 50 100 r \^ -6 + 2977 +0.12226 -1* +0.11872 + .32925 -2 + .1*0971* + .1*1829 + .75313 + .121* 3U 2 - .1*2313 - .51561 h + 9121* + .28171* 6 - 1122 - 7755 8 + 92 + 1351* 10 5 166 12 + 15 11* + 1 I6(l) +0.95890 +0.80329 X6(0) - .38251* - .39599 n = 7 n = 8 ^' r \, 100 25 100 -S + 7 + 1719 -6 + 6383 + 131 + 7722 -1* +0.25861 + 1816 +0.23556 -2 + .52093 +0.17929 + .51621 + .38995 + .961*81 + .1191*514. 2 - .58698 - .18035 - .55330 h + .21*863 + 11*76 + .20286 6 - .05770 72 - 1*203 8 + 880 + 2 + 581 10 96 59 12 + 8 5 11* 1 In(l) +0.96695 +0.99830 +0.95851 X„(0) + .28617 + .31*251 Xn(0)' -1.36551 n = 9 n = 10 n = 12 1 \-^2 r X. 100 r ^v 25 100 r \, 25 -8 -6 -h -2 2 1* 6 8 10 12 + 879 + 5205 +0.1951*2 + .50325 + .591*10 - .52913 + .17076 - 3192 + 1*01* 37 + 3 -10 -8 -6 -1* 2 k 6 8 10 12 + + 6 69 + 1189 +0.11*651 + .97729 - .11*696 + 995 1*1 + 1 + 107 + 769 + 1*037 +0.16730 + .1*8617 + .67076 - .501*15 + .-.1*599 - 2U92 + 291 25 + 2 -10 -8 -6 _u -2 2 h 6 8 + + 2 + UO + 832 +0.12367 + .981*08 - .12390 + 715 26 + 1 Xgd) Xg(0)' +0.97926 +1.99652 iio(i) Xio(O) +0.99937 + .2531*3 +0.996itu - .28280 Xi2(0) +0.99965 + .23021* 102 mCA TM No. 122ij- 11.4.- Course of the Curves X= X (7) for Low Values of the Index n ADDENDUM WACA TM 1224 LAME'S WAVE FUITCTIONS OF THE ELLIPSOID OF REVOLUTION By J. Meixner April 1949 It has recently been brought to the attention of the NACA by Miss Gertrude Blanch of the Bureau of Stsmdards, Department of Conunerce that errors exist in the tabulated values appearing in tables 11 to 17 of TM 1224. Miss Blanch notes that C. J. Bouwkamp, from whom Meixner obtained the values presented, subsequently corrected them in tables appearing in the Journal of Mathematics and Physics, vol. XXVI, no. 2, July 1947, pp. 88-91, In spite of the difference in symbols and notation in the two papers, reprints of tables I to IX included in the July 1947 issue of the Journal of Mathematics and Physics are attached for the use of those interested in receiving them. The NACA wishes to express its appreci- ation to -the Journal of Mathematics and Physics for permitting these tables to be reproduced for this purpose. Addendum to NACA TM 1221; TABLE I k' Ao Ai A. -10 2.305040 7.285254 11.790394 -9 2.130732 6.820S8S 11.192939 -8 1.959207 6.342739 10.594773 -7 1.771184 5.850492 9.997253 -6 1.571156 5.343904 9.401958 -5 1.357357 4.822809 8.810735 -4 1.127734 4.287129 8.225713 -3 0.879934 3.73C870 7.649318 -2 0.611314 3.172128 7.084258 -1 0.319000 2.593085 6.533473 2 6 1 -0.348602 1.393206 5.486800 2 -0.729392 0.773098 4.996484 3 -1.144328 0.140119 4.531027 4 -1.594493 -0.505244 4.091509 5 -2.079934 -1.162478 3.677958 -2.5996G8 -1.831051 3.289357 7 -3.151841 -2.510421 2.923796 8 -3.733982 -3.200049 2.578732 fl -4.343293 -3.899400 2.251269 10 -4.976890 -4.607952 1.93S120 TABLIO II k' Ai A. A. A. 12 20 30 42 1 11.492121 19.495277 29.496855 41.497757 2 10.990438 18.994079 28.995904 40.997089 3 10.494513 18.496.395 28.497321 40.497988 4 10.003861 18.002228 28.000923 40.000158 5 9.517981 17.511597 27.506703 39.504497 6 9.030338 17.024541 27.014846 39.010106 7 8.558395 16.541109 20.525161 38.517282 8 8.083015 16.001383 20.037710 38.026027 9 7.611465 15..'-.S5448 25.552188 37.53(1330 10 7.141428 15.11;M24 25.009492 37.048221 Addendxim to NACA TM 12 2U TABLE III 00 Characteristic function Xo{0 = 22 ^*" ^2n(?) k' 6a 6j 6. b. h bio -10 0.944709 -0.728578 0.110455 -0.007498 0.000288 -0.000007 -9 0.951472 -0.684479 0.094690 -0.005827 0.000203 -0.000005 -8 0.958380 -0.035639 0.079247 -0.004365 0.000136 -0.000003 -7 0.965363 -0.581441 0.064298 -0.003120 0.000085 -0.000001 -6 0.972311 -0.521212 0.050067 -0.002097 0.000049 -0.000001 -5 0.979071 -0.454254 0.036840 -0.001294 0.000025 -4 0.985428 -0.379882 0.02495S -0.000706 0.000011 -3 0.991099 -0.297493 0.014835 -0.000316 0.000004 -2 0.995716 -0.206682 0.006949 -0.000100 0.000001 -1 0.998846 -0.107374 0.001824 -0.000013 1 .000000 1 0.99S691 0.114368 0.001976 0.000014 2 0.994509 0.233927 0.008138 0.000118 0.000001 3 0.987210 0.350205 0.018683 0.000408 0.000005 4 0.976790 0.478301 0.033563 0.000979 0.000016 5 0.963507 0.597278 0.052483 0.001914' 0.000039 G 0.947848 0.710493 0.074931 0.003279 0.000079 0.000001 7 0.930440 0.815971 0.100266 0.005114 0.000143 0.000003 8 0.911948 0.912502 0.127799 0.007438 0.000238 0.000005 9 0.892980 0.999612 0.156881 0.010250 0.000309 0.000008 10 0.874065 1 .077435 0.180946 0.013533 0.000540 0.000014 TABLE IV Characlcrislic function .\'i(^) = ^J ^JHn/'in+iC?) k' 6i b. bt bi bt bn -10 0.964429 -0.402104 0.W6184 -0.002528 0.000081 -0.000002 -9 0.970923 -0.3G443G 0.037690 -0.001858 0.000054 -0.000001 -8 0.970877 -0.325710 0.029954 -0.001312 0.000034 -0.000001 -7 0.982232 -0.280082 0.023016 -0.000882 0.000020 -6 0.986936 -0.245730 0.016934 -0.000550 0.000011 -5 0.990948 -0.204851 0.011752 -0.000.322 0.000005 -4 0.991236 -0.163056 0.007499 -0.000164 0.000002 -3 0.996784 -0.122359 0.004197 -0.000069 0.000001 -2 0.998586 -0.081179 0.001852 -0.000020 -1 0.999651 -0.0(0326 0. 0001. TO -0.000002 1 .000000 1 0.999664 0.039616 0.000147 0.000002 2 0.998683 0.078302 0.001764 0.000019 3 0.997105 0.11009S 0.003902 0.000063 0.000001 4 0.994984 0.152711 0.006812 0.000147 0.000002 5 0.992373 0.188112 0.010436 0.000281 0.000004 C 0.989330 0.2222.30 0.014716 0.000473 0.000009 7 0.985910 0.255039 0.019595 0.000733 0.000010 8 0.982 I 67 0.2S(;500 0.025011 0.001066 0.000027 9 0.978150 0.316012 0.030908 0.001477 0.000042 0.000001 10 0.97390S 0.345380 0.037230 0.001969 0.000062 0.000001 Addendum to NACA TM 122l| TABLE V 00 Characterislic funclion Xi{^) = / . binPsnJO TAn[,K VI 00 Characteristic function Xii^) = ^ 62a+i^2n+i(£) TABLE VII 00 Charactr.ri.slic funcli2.y\»(s *' »0 h 6. b. bt bio 1.000000 1 -0.022875 0.998525 0.024445 0.000200 0.000001 2 -0.046799 0.993846 0.048736 0.000821 0.000007 3 -0.071286 0.985722 0.072766 0.001840 0.000024 0.000001 4 -0.095772 0.974150 0.096431 0.003258 0.000057 0.000001 5 -0.119671 0.959391 0.119654 0.005067 0.000110 0.000002 6 -0.142464 0.941931 0.142398 0.007260 O.OOOlitO 0.000003 7 -0.163759 0.922394 0.164677 0.009837 0.000302 0.000006 8 -0.183310 0.901438 0.186545 0.012799 0.000450 0.000010 9 -0.201017 0.879661 0.208098 0.016150 0.000640 0.000016 10 -0.216892 0.857550 0.229438 0.019902 0.000880 0.000024 k^ 4i b, 6. b, h 6.1 1 .000000 1 -0.010979 0.999565 0.017026 0.000118 2 -0.033587 0.998287 0.035224 0.000470 0.000003 3 -0.049768 0,990217 0.052770 0.001057 0.000012 4 -0.065475 0.993400 0.070253 0.001877 0.000027 5 -0.080671 0.989910 0.087061 0.002929 O.OOOO.W 6 -0.095328 0.9&';7S6 0. 10497!) 0.004212 0.000092 0.000001 7 -0.109432 0.981091 0.122202 0.005725 0.000145 0.000002 8 -0.122970 0.975879 0.I393I6 0.007465 0.000217 0.000004 9 -0.135939 0.970201 0.156318 0.009433 0.000308 0.000006 10 -0.148343 0.964106 0.173200 0.011620 0. 000122 0.000010 k' io ii b. - t. 6. bio i» 1 .000000 1 0.000091 -0.013588 0.99976S 0.013773 0.000076 2 0.000368 -0.027140 0.999071 0.027528 0.0U0304 0.000002 3 0.000834 -0.010053 0.997918 0.041266 0.000681 0.000006 4 0.001493 -0.054128 0.99U300 0.054977 0.001215 0.000015 5 0.002348 -0.067563 0.994218 0.068051 0.001897 0.000U29 0.003404 -0.080957 0.991669 0.082279 0.002729 0.000051 0.000001 7 0.004663 -0.094312 0.9886 IS 0.095855 0.003712 O.OOOO.SO O.OOfHlOl 8 0.000125 -0.107546 0.98440S 0.1092S6 0.001839 0.000120 0.000002 9 0.007806 -0.120900 0.981162 0.122811 n.0U6122 0.0(10171 0.000003 10 0.009695 -0.134130 0.976680 0.136173 0.007550 0.000231 0.000005 Addendum to NACA TM 122h TABLE VIII 00 Characteristic function XbiO = 2^ ^2n^\ Pzn+i i& k' h b. h h b, ill b„ 1.000000 1 0.000068 -0.011218 0.999854 0.011294 0.000046 2 0.000273 -0.022423 0.999418 0.022584 0.000213 0.000001 3 0.000611 -0.033609 0.998690 0.033863 0.000478 0.000004 4 0.001080 -0.044772 0.997673 0.045127 0.000850 0.000009 5 0.001681 -0.055906 0.996367 0.056373 0.001327 0.000018 6 0.002408 -0.067006 0.994772 0.067596 0.001910 0.000031 7 0.003262 -0.078066 0.992888 0.078793 0.002.598 0.000050 0.000001 8 0.004236 -0.089080 0.990719 0.089958 0.003391 0.000073 0.000001 9 0.005331 -0.100043 0.988203 0.101088 0.004290 0.000105 0.000002 10 0-.006543 -0.110949 0.985522 0.112180 0.005292 0.000143 0.000003 TABLE IX Characlcrislic function Xe({) = 2^ ^S" ^21(1) A' J. b. b. bt h bi. »ii Ju 1.000000 1 0.000050 -0.009535 0.999899 0.009571 0.000039 2 -0.000001 0.000202 -0.019061 0.999597 0.019140 0.000157 0.000001 3 -0.000004 0.000454 -0.028580 0.999094 0.028702 0.000353 0.000003 4 -0.000010 0.000805 -0.03.S087 0.998390 0.038257 0.000627 0.000006 5 -0.000020 0.001256 -0.047578 0.997486 0.047802 0.000980 0.000012 6 -0.000035 0.001807 -0.057050 0.996381 0.057331 0.001410 0.000020 7 -0.000056 0.0024-58 -0.066501 0.995076 0.006845 0.001919 0.000032 8 -0.000084 0.003207 -0.075928 0.993572 0.076341 0.002505 0.000048 0.000001 9 -0.000120 0.004055 -0.085326 0.991867 0.085816 0.003169 0.000068 0.000001 10 -0.000165 0.005001 -0.094694 0.989966 0.095268 0.003910 0.000094 0.000002 NArA-T.ancriou _ fi-in_Rn _ onn UNIVERSITY OF FLORIDA 3 1262 08105 8272