0^^i''^}^!^ 
 
 
 
AN EXTENSION OF THE 
 STURM-LIOUVILLE EXPANSION 
 
 // 
 
 / 
 
 By 
 CHESTER CLAREMONT CAMP 
 
 Respectfully submitted to the Faculty of the Graduate School 
 
 of Cornell University in partial satisfaction of the 
 
 requirements for the degree of 
 
 Doctor of Philosophy 
 
AN EXTENSION Ol^ THE 
 STURM-LIOUVILLE EXPANSION 
 
 By ^ '' 
 
 CHESTER CLAREMONT CAMP 
 
 Respectfully submitted to the Faculty of the Graduate School 
 
 of Cornell University in partial satisfaction of the 
 
 requirements for the degree of 
 
 Doctor of Philosophy 
 
ACXCHANQ^ 
 
Reprinted from The American Journal of Ma-bhkm.^tics, ypJ.XLIX, No. 3, Januarj', 1922 
 
 AN EXTENSION OF THE STURM-LIOUVILLE EXPANSION. 
 
 By Chester Cl.\remont Camp. 
 
 Introduction. 
 In 1836-7 C. Sturm's formal development of a more or less arbitrary 
 function j{x) in terms of solutions of the self-adjoint equation 
 
 d / du\ 
 TxVTx) 
 
 du 
 
 and the Sturmian boundary conditions 
 
 + 
 + 
 
 9^0 
 9^0 
 
 au{a) + a'u'(a) = 0, 
 fiu{b) + /3'i/'(6) = 0, 
 
 was considered by J. Liouville,* who undertook the problem of showing 
 that the series converges and that its value is fix). His work is important 
 although it did not satisfy all the requirements of modern mathematical 
 rigor, but in two remarkable papers A. Kneserf completely settled all the 
 more fundamental questions concerning the development. It remained 
 for HaarJ several years later to give the solution finality. 
 
 In 1908 Birkhoff |1 extended the theory not only to equations of the 
 7ith order of the form 
 
 but to systems no longer self-adjoint and conditions no longer Sturmian. 
 
 The theory is capable of extension in several directions. B6cher§ 
 considered a system of two homogeneous linear differential equations of the 
 first order and studied the roots of a solution without regard to boundary 
 conditions. Schlesinger^ took a system of n linear equations of the first 
 order with coefficients which are rational in x and obtained the asymptotic 
 forms for a solution. 
 
 The object of this paper is to discuss an extension of a problem recently 
 considered by Professor Hurwitz, namely the simultaneous expansion of two 
 
 * Liomnlle's Journal, Vol. 1 (1836), p. 253; Vol. 2 (1837>, p. 16 and p. 418. 
 
 t Malh. Ann., Vol. 58, p. 81 and Vol. 60, p. 402. 
 
 t Goettingen dissertation (1909) reprinted in Malh. Ann., Vol. 09 (1910), p. 331. Also 
 a second paper, Mcdh. Ann., Vol. 71 (1911), p. 38. See also Mercer, Phil. Trans., Vol. 211 
 (1911), p. 111. 
 
 II Trans. Amer. Malh. Soc., Vol. 9, p. 373. 
 
 § Trans. Amer. Math. Soc., Vol. 3 (1902), p. 196. 
 
 % Math. Ann., Vol. 63 (1907), p. 277. 
 
 SOTToG 
 
26 .Cave* o4?i Extmsiprupf%e Sturm-Liouville Expansion. 
 
 functions /(a;), g{x) in terms of solutions of the system 
 
 du J- -. 
 
 ^ = [X + a{x)2v, 
 
 ^ = - [X + bix)']u, 
 
 and the boundary conditions 
 
 au(a) + fiv{a) = 0, 
 a'u(b) + ^'v{b) = 0. 
 
 Although this system is simple, it is not self-adjoint and the functions 
 a{x), b{x) are any two real functions which possess continuous second 
 derivatives. 
 
 The extension I have made is that resulting from the substitution of the 
 more general linearly independent boundary conditions 
 
 aiu{a) + i3i«(a) + yiu(b) + ^iv(b) = 0, 
 a2u(a) + ^2v(a) + 72^(6) + 82v(b) = 0, 
 
 in which the coeflficients are such as to satisfy 
 
 Oil /3i 
 (X2 1S2 
 
 7i 5i 
 
 72 ^2 
 
 7^0. 
 
 I wish to acknowledge my indebtedness to Professor Hurwitz for his 
 constant interest and careful direction as well as for his recent article as a 
 basis for research. 
 
 For further references the reader is pointed to the following by Bocher: 
 Encyklopsedie der Math. Wiss., Band II, Heft 4, pp. 437-463; Proceedings 
 of the Fifth International Congress of Mathematicians, Vol. I (1912), 
 pp. 163-195. 
 
 Section I. Preliminary Lemmas. 
 
 Consider the system 
 
 Ni(uv\) = u'(x) - [X + a{x)2v(x) = 0, 
 N2{uv\) ^ v\x) + [X + b(x)2u{x) = 0, 
 
 (1) 
 
 with the boundary conditions 
 
 (2) 
 in which X is real or complex; a{x), b{x) are real functions possessing con- 
 
 Ui(uv) = aiuiO) + PMO) + 7MI) + 5iKl) = 0, 
 U2{uv) = a2u{0) + /32KO) + 72W(1) + 82V{1) = 0, 
 
Camp: An Extension of the Sturm'LioupiUe.Expansiim.'. 
 
 27 
 
 
 ai /3i 7i 5i 
 
 
 0C2 02 72 ^2 
 
 is of rank two and 
 
 
 
 (ocfi) = (76) 9^ 
 
 where 
 
 
 / 
 
 "2 P2 ; 
 
 tinuous second derivatives for ^ x ^ 1 ; and the coefficients of u, v, in 
 (2) are real constants such that the matrix* 
 
 (3) 
 
 The sohition u{x) = t(x) = is called the trivial solution of (1), (2). 
 All other solutions are termed non-trivial. 
 
 Studying non-trivial solutions of (1), (2) we derive the following lemmas. 
 Lemma I : // (um, Vm) and {iin, Vn) are two solutions of 
 
 then 
 
 Ni(ukVk\k) = NiiukVkXk) = 0, 
 
 [,'Um(x)Vn(x) — Vn{x)Vm{x)'y^=o 
 
 = (X„ 
 By hypothesis 
 
 K) J'o'[Um{s)Un(s) + »m(«)»n(*)]^J?. (4) 
 
 Wm(a^) — D^ + a(x)']i}m(x) = 0, 
 
 i?m(a*) + [Xm + b{x)2um{x) = 0, 
 
 u'„(x) — [X„ + a{x)2 Vn{x) = 0, 
 
 ' v:(x) + lK + b(x)'}u^ix) = 0. 
 
 If we multiply these equations by Vn{x), — «„(.r), — Vm{x), and Um(x) 
 respectively, add, and integrate from to x, we obtain the equation (4) 
 required. 
 
 Lemma II: // (u, v) satisfies Ni = <p, N2 = yp, where (p, \J/ are functions 
 of X continuous ^ a: ^ 1, and (U, V) satisfies Ni = N2 = for the same 
 value of X, then 
 
 \:u{x)V(x) - Uix)v(x)yo = Jl'LV(s)<p(s) 
 By h}T)othesis 
 
 U{s)rl^{s)']ds. 
 
 (5) 
 
 u'(x) - [X + a(x)-]v(x) = <p{x), 
 v'{x) + [X + h{x)'}u{x) = yl^{x), 
 V'{x) - [X + a{x)']V{x) = 0, 
 V'{x) + [X + h{x)']U{x) = 0. 
 
 • If the matrix is of rank less than two the conditions (2) are not independent. The 
 condition (o^) = (78) is somewhat analogous to the condition of self-adjointness for single 
 equations of order two. The case in which {(x0) = (76) = can obviously be reduced to 
 the problem considered by Professor W. A. HiuTvitz. 
 
28 .Caiwp: ^n Extension of the Sturm-Liouville Expansion. 
 
 Multiplying respectively by V(x), — U(x), — v{x), u(x); adding and 
 integrating as before we get equation (5) above. 
 
 Lemma III: Two pairs of functions u{x), ti{x)', U(x), V(x); satisfying 
 Ui = 1/2 = 0, satisfy also the relation 
 
 [_u{x)V(x) - v(x)U(x)J = 
 when (3) is satisfied. 
 We are given 
 
 «iw(0) 4- ^iHO) + Tiw(l) + 511^(1) = 0, 
 
 a2u{0) + /32KO) + 72W(1) + 521^(1) = 0, 
 
 aiU{0) + ^iF(O) + yrU{l) + 5iF(l) = 0, 
 
 a2U{0) + /32F(0) + 72U{1) + 62F(1) = 0. 
 
 Since (aj8) 9^ 0, 
 
 1 l7iM(l) + Wl), /3i 
 
 (6) 
 
 w(0) = - 
 
 and 
 
 v{0) = - 
 
 (afi) \ T2W(1) + ^2^1); ^2 
 1 ! ai, Tiw(l) + 8iv{l) 
 
 {al3)\a2, 72^(1)4-52^1) 
 
 also 17(0), F(0) are similar functions of ^7(1), F(l). 
 
 Substituting these values in (6) we get for the condition necessary that 
 
 (a/5)2 = (5/3) (7a) + (7^)(a5) = {a^){y8) 
 
 which is evidently true by (3). 
 
 Lemma IV: // (ui, vi), (u2, V2) are two linearly independent pairs of 
 functions of x, continuous = x = 1, then 
 
 G2^ 
 
 (11) (12) 
 (21) (22) 
 
 is real and greater than zero, where 
 
 (rs) ^ yoTwr(«)ws(*) + Vr{s)vs(s)']ds 
 
 and u{x) 
 
 represents the conjugate of 
 
 u{x). 
 
 
 Obviously G2 = 
 
 -G2, 
 
 i.e., G2 is real. 
 
 Let 
 
 
 
 
 
 
 
 p{x) = 
 
 (11) ur{x) 
 (21) U2{x) 
 
 J 
 
 
 
 
 q{x) ^ 
 
 (11) ^i{x) 
 (21) t2{x) 
 
 } 
 
 i.e., 
 
 
 
 p{x) = CiWi(.r) + C2W2(a;), 
 
 
 
 
 q{x) = ci 
 
 ^A^) +,C2»2(i 
 
 .). 
 
Camp: An Extension of the S^t7mrLiouviUe.E::tpp4i^on/ 29 
 
 Calculate 
 
 ^,r / N- / X . / N / x-ij 1(11) (1^)1 {0,i{k=l, 
 fo^Lp(x)u.(^) + ry(.)..(.)]rf. - I ^2i; l^k) r 1 (?„ if ^ = 2. 
 
 Then 
 
 fo'i\pi^)\'+ \q(x)\')dx^ fo'\j>i^)p(^) + Q(^Vjix)y^ 
 
 = Cifo'\JP(x)Mx) + q{x)vi(x)ya- + royo'Qj(.r)M2(.T) + q{z)v2{x)yx 
 = C2G2 ^ 0. 
 
 If (?2 = and C2 = P2 7^ 0, then p(ar) = g(a;) = and (wi, rO, (1*2, ^'2) 
 will be linearly dependent. 
 
 If 6*2 = and Cz = 0, then since 
 
 C2 ^ (11) ^ yoHki(a:)|^+ |^i(a:)|2)rfa: = 0, 
 
 Wi = n = and they will again be linearly dependent, thus violating the 
 hypothesis. 
 Hence 
 
 (?2 > (7) 
 
 since Co > 0, and the lemma is proved.* 
 Lemma V: // (3) is satisfied, then 
 
 [(76) + (a/3) J ^ l(ay) + W8)J + [(a5) + (7^) J. (8) 
 
 (8) can easily be shown to be equivalent to 
 
 , (78)2 + (afi? ^ ("7)' + W^y + (ccdy + (7/3)^ (9) 
 
 But • 
 
 (a7)2+(W^2(a7)(/98), 
 and 
 
 Hence the right member of (9) is 
 
 ^ 2(ay)(fi8) - {a8)(^y) 
 ^ 2(a^)(76), or by (3) 
 ^ {a^y + (75)^ 
 
 , Since (9) is true, (8) is also. 
 
 ' Lemma VI: // (3) holds, it is impossible for the coefficients in (2) to satisfy 
 
 {ay) + m) = 0, 
 (a5)+(7^) = 0. ^ 
 
 •The lemma admits of the following generalization: If (ui, Vi), (uj, Vt), ••• («„, f«) 
 are n linearly independent pairs of functions of x, continuous ^ 2 ^ 1, then 
 
 M (11) (12) ••• (In) 
 ^ _ : (21) (22) • . • (2n) 
 
 I (nl) (n2) • • • (nn) \ 
 is real and greater than zero, n being a positive integer. 
 
30 ^AM^* <^^ EgdeV'Siqn of the Sturm-Liouville Expansion. 
 
 By (3) 
 
 (a^) - (75) = 0. 
 
 If then we assume (10) true, by squaring and adding all these equations, 
 we have 
 
 (ayy + m' + (a8y + (7/3)^ + (a^Y + {y8y + 2(^7) (^S) 
 
 + 2(a8)(7/3) - 2(«^)(75) = 0. 
 
 The sum of the last three terms of the first member vanishes identically 
 so that if (10) holds, each of the six determinants must vanish. Since 
 this violates (3) the proof is complete. 
 
 Section II. Properties of Solutions of the Homogeneous and Non-homo- 
 geneous Systems. 
 Lemma I: A necessary and sufficient condition for the existence of a 
 solution {u, v) of (1), (2) is that the determinant 
 
 D{\)^ 
 
 UiiUiVi) Ui{u-iV2) 
 UiiUiVi) U2{llllV2) 
 
 (11) 
 
 vanish for some value of X, where (ui, vi), {ui, V2) are solutions of (1) defined by 
 
 um = 1, vM = 0, 
 
 W2(0) = 0, i)2(0) =1. ^ ^ 
 
 By the existence theorem we know that a solution either of (1) or of the 
 corresponding non-homogeneous system 
 
 Ni = <p, N2 = rp, 
 
 where <p, x}/ are functions of x, continuous ^ a; ^ 1, will be entire in X, 
 provided it is defined by 
 
 'w(O) = ci, 
 v(0) = Co, 
 
 in which the c's are independent of X. Clearly D{\) will also be entire in X. 
 Any solution of (1) is expressible in the form 
 
 u(x) = u(0)ui(x) + v(0)u^{x), ,.„. 
 
 v{x) = u{0)vi(x) + v{0)v2{x). 
 
 In order that u(x), v(x) satisfy (2) it is necessary and sufiicient that u(0), 
 v{0) be determined so as to satisfy 
 
 Ui{u) = Ui(UiV,MO) + Ui(U2V2)v(0) = 0, ,j^. 
 
 U2{U) = U2(UiVi)u(0) + U2{U2V2)V(0) = 0, 
 
 or if u{x), v(x) is to be a non-trivial solution, the determinant of coefiicients 
 of w(0), v(0), which is P(X), must vanish. This is obviously also sufficient. 
 
Camp: An Extension of the Sturm-Liouville Expajision.^ 31 
 
 Lemma II: The roots of D{\) are all real and the solutions of (1), (2) 
 may he taken as real. 
 
 IjQt X = Xjt be a root of D{\). Then (uk,Vk) satisfies Ni(ukVk\k) =_0, 
 NiiukV^y^k) — and {uk, Vk) will satisfy the same conditions for X = Xjt. 
 Hence by Lemma I, Section I 
 
 {uk'Ck — UkV)P^» = (Kk — \k)f'Ouk{s) |2 + \vk(s) \^)ds. 
 
 Moreover {uk, Vk) satisfies Ui(u) = U-i{u) = so that by Lemma III, 
 Section I _ 
 
 (Xt-x,)y;Hk*|2+ \vk\'')dx = {). 
 
 If Xjt 7»^ Xfc, {uk, tk) is a trivial solution which we have excluded. Therefore, 
 Xjfc = Xfc and Xjt is real. 
 
 Since each equation of the system (1), (2) is linear and a{x), b{x) are 
 real, any solution of the system if complex may be broken up into real and 
 imaginary parts which will separately satisfy the same system. When 
 D{\) vanishes the ratio of u{0) to i'(0) or of v{0) to m(0) is determinable* and 
 by the existence theorem the solution u(x), v{x) is then uniquely determined 
 except for a constant factor. Hence the coefficients of the real and imag- 
 inary parts constitute two solutions which are linearly dependent. It is 
 therefore clear that we may restrict ourselves to real solutions of the system. 
 
 A similar discussion will show that we need consider only real solutions 
 
 of the system 
 
 Ni = <p, N2 = ^, (15) 
 
 where (p, \f/ are real functions of x and X is restricted to real values, a restric- 
 tion which we shall henceforth make. 
 
 A value of X which makes Z)(X) vanish is called a principal parameter 
 valve and the corresponding solution of the homogeneous system (1), (2) 
 is known as a principal solution. 
 
 Since D{\) is an entire function it cannot have more than a finite 
 number of roots in any finite interval of the X-axis, so that we may designate 
 the roots of D{\) by \„, n = 0, ± I, ± 2, ■■■. 
 
 Theorem I : A sufficient condition for the existence of a solution (m, v) of 
 (15), (2) analytic for all real finite values of\is that 
 
 '1""m t<1k = (16) 
 
 X 
 
 for every (w„, Vn) satisfying (1), (2) /or X = X„, n = 0, zb 1, ± 2, • • •. 
 We shall consider in turn three kinds of values of X: 
 Case I: Those for which the matrix of (11) is of rank two; 
 
 * If the matrix of (11) is of rank zero (Case III below), then this is found by the eval- 
 uation of an indeterminate form. 
 
32 Camp I An Extension, of- the Sturm-Liouville Expansion. 
 
 Case II : Those for which the matrix is of rank one (at X = X„) ; 
 Case III: Those for which it is of rank zero (at X = X^). 
 Take a solution of (1), 
 
 U(x) = CiUi(x) + C2U2(x), 
 V{x) = CiViix) + C2V2(x), 
 
 which is identical with (13), provided w(0) = ci, v(0) = c^. This solution 
 will satisfy (2) if (14) is satisfied and will be non-trivial provided Ci, c^ are 
 not both zero. For Case I at least two of the elements of Z)(X) must be 
 different from zero. Without loss of generality assume that f/i(?^2«2) 9^ 0, 
 then by continuity it will not vanish for values of X nearby. Put 
 Ci = — Ui(u2V2), C2 = Ui{uivi). Then Uiiuv) = 0, and U2iuv) = D{\). 
 Define another solution (C/, F) of (1) by 
 
 U(x) = — C2Ui(x) + CiU2ix), 
 V{x) = — C2Vi(x) -h CiV2(x), 
 
 Then (u, v), {U, V) will be linearly independent since 
 
 (18) 
 
 W = 
 
 liiO) <0) _ c, C2 2 , 2 ^ n 
 
 — = C] + Cj > 0. 
 
 — C2 Ci 
 
 U(0) F(0) 
 
 Consider a solution {u, v) of (15) defined by 
 
 u(x) = wo(a-) + biu{x) + b2U(x), 
 v(x) = vo{x) + biv(x) + 62^(0-), 
 
 where (uq, vq) is a particular solution of (15) such that 
 
 Uo{0) = i'o(O) = 0. 
 Then 
 
 1 U2{uv) = U2(.uoVo) + b,U2(vv) + bolMUV), 
 and (u, v) will satisfy (2) if 
 
 (19) 
 
 (20) 
 
 62 = 
 
 &i= 
 
 W ' 
 
 Ul (UoVo)U 2 iUV) - U2( UoVo)W 
 WU2(UV) 
 
 (21) 
 
 , , , U,(UoVo)U{x) Ui(UoVo)U2(UV) + U2(UoVo)W ^ ^ 
 
 u{x) = u^ix) H ^ ^^^^^ u{x), 
 
 ^ ^ , Ui(UoVo)V{x) L\{UoVo)U2iUV) + U2 iUoVo)W ^^^^ 
 
 v(x) = Vo{x) -\ ^ ^^^^^^ v{x), 
 
 since U2{uv) = D(k). 
 
 Hence if the matrix of (11) is of rank two, (22) determines a solution 
 
Camp: An Extension of the Sturm-Liouville Expansion. 
 
 33 
 
 ^, v) analytic in X for that range for which Ui{u2V2) 9^ 0. For other ranges 
 of vklues of X within which U\{u-yV2) vanishes we choose in its place another 
 element of the determinant P(X) which differs from zero throughout that 
 range. This process determines h\, hi in a different fashion. Clearly since 
 the roots of UiiihVi) and of every other element of D{\) are isolated, each 
 element being entire in X, the solution {u, v) determined in either way" for 
 points of the X-axis at which UiiihVi) 9^ and some other element, e.g., 
 UiiuiVi) 7^ 0, will be identical since there is one and only one such solution 
 by the existence theorem. .*. The proof is complete for Case I. 
 
 Case II: Let us first show that no root of Z)(X) is double for this case. 
 Assuming that Uiiv^ih) 5»^ for X = Xn and defining Ci, Cj as before, we have 
 by differentiating as to X, since u{x), v{x) of (17) are entire in X and 
 
 satisfy (1): 
 
 [Niiv^vO^) = vix), 
 
 [Niiu^vM^ -u{x). ^^^^ 
 
 If Z)(X) has a double root X = X„, then for this value of X 
 
 Uiinv) = 0, C/2(wx«x) = 0; Ni(uv\) = NiiuvX) = 0; 
 Uiiuv) = 0, Uiiu^v^) = 0. 
 
 Whence by Lemmas II, III, Section I 
 
 fo%'u{s)J-\-[v{s)J)ds^O. 
 
 This is impossible since w(0) = — UiiUiVi) ^ 0. Therefore Z)(X) has no 
 double root.; 
 
 For values sufficiently near Xn, Z)(X) 7^ and the solution (w, i) of (15), 
 (2) analytic in X is given by (22). And by Lemma II, Section I 
 
 luo(x)v(x) - u(x)vo(x)']^ = fo'[.v(s)<p(s) - uis)xf^{s)ys, (24) 
 
 liK^ixWix) - U(xMx)J, = fo'lV(s)<p(s) - U{s)^P{s)-]ds. (25) 
 
 The left members vanish at the lower limit since Uo(0) and Do(0) are equal 
 to zero, hence solving we obtain 
 
 u(x) 
 u{s) 
 vis) 
 
 v(x) 
 u(s) 
 vis) 
 
 X„, w(ar) approaches the value 
 
 iMiix) 
 
 voix) = 
 
 '4: 
 
 Uix) 
 Uis) 
 Vis) 
 
 Vix) 
 Uis) 
 Vis) 
 
 I 
 
 <pis) \ds, 
 
 ^(*)l 
 
 
 
 <pis) ds. 
 
 rPis) 
 
 (26) 
 
 (27) 
 
 If X 
 
 .^(,)+?^C,(,)--^) 
 
 ^LUiilM>Vo)U2iUV) + U2iUoVo)W2 
 
 2>'(X) 
 
 (28) 
 
 *=*, 
 
34 Camp: An Extension of the Sturm^Liouville Expansion. 
 
 provided only 
 
 UiiuoVoW^iUV) + U2(uoVo)W:\,^,^ = • (29) 
 
 since Z)'(X) 9^ 0. 
 
 Again if (29) holds, vix) will approach a limit as X -► X„. In such a 
 case (u, v) will be continuous if put equal to the limit approached at X = Xn 
 and therefore analytic for all X in the interval. 
 
 Since W= - UiiUV), from (26), (27) it follows that (29) is equivalent to 
 
 £ 
 
 
 
 TiC^(l)+5iF(l), UriUV) 
 72t/(l)+52F(l), U^iUV) 
 u{s) U(s) <p(s) 
 
 <S) Vis) ^(*):x=Xn 
 
 7iw(l)+5i2)(l), UiiUV) 
 
 T2W(1) + 52<1), U2iUV) 
 
 ds=0. 
 
 n 
 
 (30) 
 
 The first element of the large determinant is the same as 
 
 I 7iw(l) + 5iKl), yiUil) + 5iF(l) 
 
 I 72W(1) + 52^1), 72^^(1) + 52^(1) 
 
 aiw(O) + fiMO), aiUiO) + /3iF(0) 
 «2W(0) + i82K0), a^UiO) + i82F(0) 
 
 since for X = X„, Ui(uv) = Uziuv) = 0, or simplifying further, this element 
 reduces to (y8)W - (afi)W = 0. 
 
 Thus a sufiicient condition for the existence of a solution {u, v) of (15), 
 (2) analytic for all X when D{X) is of rank one is that (16) be satisfied for all 
 n such that D{\n) = 0, and the theorem is proved for Case II. 
 
 Case III. When every element of Z)(X) vanishes for some X = \k, it 
 is obvious that D'(Kk) also vanishes. Let us show that Z)"(Xjk) 7^ for 
 such a case. 
 
 By the same reasoning as was used to prove D'(X) ?^ for X = Xn in 
 Case II we show that if C7i(wix«ix) and U2{uixV\^ both vanish at X = Xt, 
 then 
 
 f^\[u,{x)J + lv,{x)J)dx = 0, 
 
 which is impossible since wi(0) = 1. Hence ?7i(wix«ix) and U2(uixVi\) can- 
 not both vanish at X = X^. Similarly ?7i(w2x^2x) and U2(u2\V2\) cannot both 
 vanish there. Without loss of generality we may assume 
 
 C7i(M2x«'2x) ^0, X = X,. (31) 
 
 Now define a solution of (1) by 
 
 u{x) = Ui{x) -h —ihi^), 
 
 ""' (32) 
 
 v(x) = Vi(x) + - Viix), 
 C2 
 
 and choose 
 
 £l_ _ UljUiVi) _ 
 C2 Ui{2l2V2) 
 
Camp: An Extension of the Sturm-Liouville Expansion. 35 
 
 For values of X sufficiently near Xjt, Ui(ihVo) 7^ 0, otherwise (31) would 
 be violated by Rolle's Theorem. Hence for all X in the interval considered 
 
 Consequently for all such X, UiiuxV^) = 0. 
 Again 
 
 At X = \k, 
 
 And 
 
 For X = \k, 
 
 D"{\) _^ D{\) C7l(W2XXl'2Xx) + D'(\) U,{u,y) 
 
 U2(UxTx) = - 
 
 Ui{'lh\V2x) 2Ui{UiV2)Ui{v^ix'02\) 
 
 This vanishes if Z>"(X) = at X = Xjt. Applying the argument a third 
 time we obtain 
 
 /oKw' + ''^)dx = 0, 
 
 which is absurd since by (32) w(0) = 1 . 
 
 Since for Case III when X = X^- every solution of (1) satisfies (2), if we 
 define a solution of (15) by 
 
 fu(x) = Vo(x) + hiiti(x) + biU^ix), .^„v 
 
 1 V{X) = Vo(x) + biViix) + b2V2(x), ' ^•^'^^ 
 
 where Wo(0) = ro(0) = defines a particular solution of (15), then 
 U\{ut) = Ui{uoVq) and U2{uv) = U2(uoVo). If these vanish fei, 62 will be 
 arbitrary. Again by using results similar to (24), (25), (26), (27) we have, 
 since 
 
 wi(0) ri(0) i _ , 
 
 «2(0) tJ2(0)| ^' 
 
 /•I 7iWi(l) + 5i»i(l), 7iw^(l) + 5x^2(1), I 
 UiiltcVo) = — I ui(s), v^{8), (p(s) ds, 
 
 U2(.lliiVo) 
 
 «/0 
 
 72Wl(l) + 52»l(l), 72W2(1) + 52^^(1), 
 
 «i(*), Ms), <p{s) 
 
 ^i(*), «2(*), i{s) 
 
 ds. 
 
 For X = Xfc, (wi, ti), (m2, t>2) are solutions of (1), (2). Hence if we assume 
 
36 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 that (16) holds for all solutions (Un, Vn) of (1), (2), then (w, v) will satisfy 
 (2) for bi, 62 arbitrary. 
 
 We wish to choose values for them such as to make the solution (w, v), 
 now analytic X 5^ Xjt, continuous at X = Xa;. 
 
 Since every solution of (1) may be expressed as a linear combination of 
 ui, Ui, vi, V2, we may put (33) in the form 
 
 fu(x) = Uq(x) + diu(x) + dzMx), ,oA\ 
 
 in which {u, v) is defined by (32) and ei/c2 has the same value as before. 
 {u, v) and (^2, ^2) are linearly independent since 
 
 w(0) v{0) 
 
 W2(0) «2(0) 
 
 1 ,Ci/c2 
 
 1 
 
 5^0. 
 
 As TJ\{uv) ^ 0, we have 
 
 Ui{uv) = Ui{uoVo) + d^tfiiu^v^), 
 which will vanish for all X in the interval considered if 
 
 Again 
 
 Uziuv) = UiiuoVo) + diUiiiiv) + diUiiuiiOi) = 0, 
 provided 
 
 7 ^ _ U2(UqVo)Ui(U2V2) + Ul(UoVo)U2(U2V2) ^ 
 Ui(U2V2)U2{uv) 
 
 For X ?^ X;fc, 
 
 Ui{U2V2) 
 
 and 
 
 J _ U2(UoVQ)Ui('WiV2) -h Ui{UoVo)U2(lhV2) 
 
 * W) 
 
 Hence for X = Xa; in a sufficiently small interval, Z)(X) 5^ and Ui(u2, V2) 9^ 0, 
 «(x) = «„W+ ^^("»'^) V^{MV.(v4oi) V.{v.^i> ^(^^ _ Mf4^(^). (35) 
 
 -L'(X) Ui{ll2V2) 
 
 If we let X approach X^ we get as the value approached by the right member: 
 
 rr r \ :SSr2 C ^2 (uqVo) Ui {U2V2) + Ui (tioVo) U2 {U2V2) ] 
 
 u,{x) - ^^i^ U2{x) + ^^(.r) ^-^ , 
 
 Ui{U2xV2x) D"(K) 
 
 X = Xit. 
 Evidently u{x) approaches a limit since 
 
 £1 _^ _ UijihxVix) _ 
 
 C2 UiiU2\V2x) 
 
Camp: An Extension of the Stnrm-Liouville Expansion. 37 
 
 Also 
 
 — [,U2('UqTo)Ui(U2V2) + Ui(UoVq)U2{'U2V2)'] =0, X = Xfc, 
 
 ah. 
 
 since each factor of both terms vanishes there. Also Z)'(Xt) = 0. 
 
 Hence u{x) approaches a limit as X -* Xjt, and similarly v{x) does also. 
 Thus we have a solution (u, i) analytic for all X in a small enough interval 
 about X = Xjk, which satisfies (2). By extending the reasoning as in Case 
 I the proof is completed. 
 
 Theorem II: If for n = 0, ± 1, ± 2, - - ■ 
 
 f 
 
 Unis) <p(s) 
 
 i^n(s) rp(s) 
 
 ds = 0, 
 
 then (p(x) = \p(x) = 0. 
 
 The proof will be merely outlined here since it is given in full in a recent 
 article by Professor W. A. Hurwitz. 
 
 Let the solution of (15), (2) shown by the previous theorem to exist be 
 represented by 
 
 u(x) = yo{x) + \yi(x) + \^y2(x) + • • •, 
 . v{x) = zoix) + \z,{x) + \'Z2{X) + . . • . ^'^^^ 
 
 Since X is restricted to real values we have but real functions of x with 
 which to deal. Putting (36) in (15), (2) and equating coefficients of X* 
 we get sets of differential equations and boundary conditions satisfied by 
 {ym, Zm), m ^ 0, 1, 2, •••. By eliminating a(x), b{x) from two different 
 sets and using Lemma III, Section I, one gets 
 
 /oiymyn + Z,nZn)dx = (JVy,n-\yn+l + Z„^lZn+l)dx = W^^ny 
 
 where 
 
 ^r = Jo^iyryo -f ZrZo)dx. 
 
 Then from the inequality 
 
 2 
 
 y„^i{x) ym^i(^) 
 
 Vmr-lix) ym^liO 
 
 + 
 
 + 
 
 y«_l(x) 2„_i(^) I 
 
 2Wfi(a:) ym+iiO 
 
 Zm^l{x) yn^iiO 
 
 + 
 
 Z„^l{x) 2W+-l(^) 
 
 ^0, (37) 
 
 for x and ^ in the interval from zero to one, by integrating as to x and ^ 
 successively from to 1 and simplifying, we have 
 
 W2n^2W2r^2 " Wl^ ^ 0. (38) 
 
 Next comes the lemma that some W^2ik = 0. Assume no ^'2* = 0. Multi- 
 plying the series of (30) by y^ix), zq{x) respectively, adding and integrating 
 
38 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 we obtain the uniformly convergent series 
 
 This converges absolutely as also does 
 Using (38), i.e. 
 
 (39) 
 (40) 
 
 forX 
 
 =v 
 
 Wo 
 
 W, 
 
 W2^Wi = We' 
 
 in this we get an absurdity 
 
 ^0 1 + \Wo\ H" I PFo I + • • • convergent. Thus the lemma is proved, 
 
 I.e. 
 
 foKyl + 4)dx = 0, or yk{x) = Zk{x) ^ 0. (41) 
 
 Thence from the differential equations 
 
 yk-i(x) = yk-2{x) = ... = yQ{x) = 0, 
 and similarly for z and also 4/(x) = (p{x) = 0. Q.e.d. 
 
 Section III. Asymptotic Formulae. 
 Theorem III: For |X| large a solution of (1) defined by u{0) = a> 
 v{0) = j8 takes the form 
 
 u{x) = a cos ^ + /3 sin ^ + I r- 1' 
 vix) = jS cos ^ — a sin ^ + I ^ 1' 
 
 (42) 
 
 and its partial derivatives, the form 
 
 where 
 
 ^x(^) = l^x cos ^ — ax sin ^ + 
 
 Vx{x) = — ax cos ^ — jSa; sin ^ + 
 
 ^ = X.T + hfo'Lais) + b(s)2ds. 
 
 & 
 
 (43) 
 
 (44) 
 
 I give an outline of a proof analogous to that used in Professor Hurwitz's 
 recent article. 
 Assume 
 
 u(x) = U+(l+ ^\ {a cos ^ + /3 sin 0, 
 v(x) = F +n + -j^)()3 cos I - a sin ^). 
 
 (45) 
 
Camp: An Extension of the Sturm-Liouville Expansion. 
 Putting in (1) we get 
 
 39 
 
 (46) 
 
 where (p, \f/, <px, ^x are 0(1) as regards X. 
 
 Using as multipliers cos "Kx, — sin \x, adding, integrating from to x> 
 and combining this with the result of repeating the procedure with sin Xx, 
 cos \x as multipliers we get 
 
 
 (47) 
 
 where F, G and the K's are 0(1), and F^, Ox, K^ are also. , 
 
 ■•• UM = f + Jl'lKnUis) + K,2V{s) + KzzUM + K^,VMlds, 
 
 (48) 
 
 in which H, J, and the K*s of (48) are 0(1). By the process of successive 
 approximations we obtain from (47), (48), 
 
 Uix) = (^^^ V(x) = 0' UM = (~j' VM = 0- (49) 
 
 From (45), (49) we see that (42) is true. Also by differentiating (45) as 
 to X and using (49) the rest of the proof is obvious. 
 
 Corollary: For |X| large Z)(X), D'{\) take the forms 
 
 Z)(X) = ^1A' + B' sin [^(1) + ^'] + C -{- (^\ 
 D'{\) = <A^ + 52 cos K(l) + ^] + (i V 
 
 where 
 
 A ^ {ay) + (i95), B = {ab) + (7/3), C = 2(a^), 
 and ^p is defined by 
 
 cos (f = 
 
 sm <p 
 
 B 
 
 (50) 
 
 (51) 
 
 (52) 
 
 By Lemma VI, Section I, A and J5 cannot both vanish. From Theorem III 
 
40 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 ui{x) = cos ^ + ( -• J , 
 
 vi(x) = — sm^ + 0(-j> 
 
 thix) = sin ^ + f - j » 
 
 V2(x) = cos ^ + ( - J • 
 
 Putting these in (11) we get 
 
 D(X) = A sin ^(1) + B cos ^(1) + M) + {y8) + o(^\ 
 
 (53) 
 
 (54) 
 
 Again using the asymptotic values for uix{x), vix{x), etc., from (53) in the 
 identity 
 
 we derive 
 
 
 + 
 
 C^2(Wix?Jlx) U2{'lh\V2)d 
 
 D'(X) = A cos ^(1) - B sin ^(1) + (^\ 
 
 and (50) follows by (3), (51), (52). 
 
 To show D{X) has roots, no matter how large |X| is, consider the 
 Theorem IV: If In is a principal parameter value for the system (1) and 
 
 the boundary conditions 
 
 jaMO) + iSiKO) = 0, 
 |Tiw(1)+ 8M1) = 0, 
 
 (55) 
 
 then there exist exactly two roots of D(\) in the intervals: 
 
 Case I. {hp, hp+i), p = 0, db 1, ± 2, • • • , if P21 > for X = /i; 
 Case II. (/2JH-1, hp+i), p = 0, ± 1, =b 2, • • •, if P21 < for X = h* 
 Define 
 
 'Bi(x) = aiu(x) + I3iv{x), 
 
 Bzix) = a2u{x) + ^2'v{x), (56) 
 
 P\{x) = — yxu{x) — 8iv(x), ; 
 
 Piix) = — y2u(x) — hvix), 
 
 and determine two solutions, [ui{x\), vi{xKy\, [u2{xk), V2{x\)'}, of (1) 
 such that 
 
 fBn(O) = 0, 52i(0) = 1, 
 
 Ui2(0) = 1, ^22(0) = 0; 
 
 (57) 
 
 in which the second subscript refers to the solution involved. Without 
 
 * I have followed the method of proof used by Professor Birkhoff in an article published 
 in the Transactions in 1909 entitled, "Existence and Oscillation Theorem for a Certain 
 Boundary Value Problem." 
 
Camp: An Extension of the Stiirm-Liouville Expansion. 41 
 
 loss of generality we may assume that the coeflScients ai, /3i, 71, 5i have 
 been divided by a constant such that 
 
 (aiS) = (76) = 1. (58) 
 
 Then we prove the 
 
 I^EMMA I: At a simple value X„ of X, Z)(X) changes sign in such a way 
 that D'{\) has the sign of — Pn or P22, where 
 
 jPn = - TiWi^l) - 5i?Ji(l), ff.Q. 
 
 IP22 ^ - 72W2(1) - 52^2(1). ^^^^ 
 
 The solutions (ui,vi), (ui,V2) defined by (57) are linearly independent since 
 
 = (afi) = - 1 (60) 
 
 wi<0) tJi(O) ! ^ - ^1 ai 
 
 W2(0) ^2(0) I ^2 - Ci2 
 
 and hence by a relation similar to Abel's 
 
 \u{x), v{x) I = constant 7^ 0. 
 Thus 
 
 Bn(x)B22(x) - Bi2(x)B2i(x) = - 1, ^ a; ^ 1. (61) 
 
 Likewise 
 
 Pn(x)P22{x) - Pnix)P2iix) = - 1, ^ a: ^ 1. (62) 
 
 Also any solution of (1) may be written 
 
 u{x) = CiWi(ar) + C2i^(a-), 
 
 V{X) = CiViix) + C2V2{X). 
 
 If this is' to satisfy (2) we must have 
 
 £>(X) = L " p'^' ^" p'H = 0. (63) 
 
 i i — i^21> "~ -t 22 I 
 
 We have shown that the necessary condition for a double value of X is 
 that Z)(X) be of rank zero, i.e., 
 
 Pn = P22 = 0, 
 P21 = Pn — 1. 
 
 (64) 
 
 This is obviously also sufficient. 
 Since 
 
 Ni(uivi\) = Niiuivik) = (65) 
 
 and 
 
 ^■l(Wlx^lxX) = vi, NiiuiyVixX) = - u, (66) 
 
 by Lemma II, Section I, 
 
 wix^i — wifix = yo'(w? + T}'i)d^- (67) 
 
 Similarly 
 
 «2x^ — ihV2x = fo'(nl + vDds. (68) 
 
42 Camp: An Extension of the Sturm- Liouville Expansion. 
 
 Again from 
 
 NiiuiViX) = NiiuiViX) = (69) 
 
 and (66) by the same Lemma 
 
 Uix'Ci — U2V1X = So'iuiUi + V1V2) ds. (70) 
 
 Similarly 
 
 U2kVi — U1V2K = J'o''iUiU2 + ViV2)ds. (71) 
 
 From (67), (70) 
 
 wu(a:) = fo'^liul + vl)u2(x) — (uiU2 + ViV2)uiix)2ds. (72) 
 
 Similarly 
 
 ^ik(x) = JTLi'^l + ^1)^2(3:) — (W1W2 + viV2)vi{x)']ds. (73) 
 
 Also from (68), (71) 
 
 ^\(a^) = JVL(uiU2 + viV2)u2{x) — (u] + vl)ui{x)']ds, (74) 
 
 V(^) = yo'C(wiW2 + «i»2)i'2(a:) — (w2 + vl)vi{x)2ds. (75) 
 
 Now from (62), (63) 
 
 D(K) ^ Pu + P21 - 2. (76) 
 
 Henceby(72), (73), (74), (75), (76) 
 
 D'(K) = Pi2x + P2U 
 
 or 
 
 D'W = fo'lP22u\ + (P12 - P2l)WlW2 - Pliwi 
 
 + P22'vl 4- (P12 — P2i)^'i^2 — Pnvr\ds. 
 
 (77) 
 
 The integrand consists of two quadratic forms whose discriminant is 
 
 (P12 - P21)' + 4PnP22 (78) 
 
 or by (62) 
 
 (P12 + P2i)^ - 4. . (79) 
 
 This vanishes when D(X) does. For a simple root of D (X)Pii, P22 cannot 
 both vanish since then by (78) and (63), (64) would be satisfied. Again 
 neither form can vanish because {ui,vi) and (^2,^2) are linearly independent. 
 Thus the Lemma is proved. 
 
 Lemma II: At a double valu£ of X, say Xa, D{\) maintains a negative sign. 
 
 From (64) 
 
 APii = Pii, AP22 = P22, 1 + AP12 = P12, 1 + AP21 = P21, (80) 
 
 in which 
 
 P= P(\-\- AX), X = X,. (81) 
 
 Then from (62) 
 
 P11P22 — P12P21 = ~ 1 
 or 
 
 AP11AP22 - (1 + APi2)(l + AP21) = - 1, (82) 
 
Camp: An Extension of the Sturm-Liouville Expansion. 43 
 
 and by (76) 
 
 AD = APi2 + AP21 = APuAP-n - AP12AP21. (83) 
 
 Clearly 
 
 APu = - yiAuiil) - 5iAri(l), 
 
 and so for the others. Also 
 
 Ni(Au, Av, X) = A\v, 
 
 NiiAu, Av, X) = — AXw. 
 
 Awi Aiiz Aci Avi 
 Using these for (?^i,»i), Cm2,»2) we get results for .^, -rr-, -rr-, -rr 
 
 which differ little from the right members of equations (72) to (75). Putting 
 these values in (83) and omitting in each term infinitesimals of order 
 greater than two, we obtain 
 
 A^ = \lfo'(uiU2 + ViV2)A\dsJ - f,Ku\ -f v\)A\dsf,\u\ + v-^AUs. 
 
 By Lemma IV, Section I, AD < 0, since the solutions are real, and 
 D{\) preserves a negative sign at X = X^. 
 
 Lemma III: Z)(X) has the same sign as P21 at the values \ = In, n = 0, 
 ± 1, rfc 2, • • •, unless P21 = 1, when D{\) = 0. 
 
 Since Wi(0) = — /3, ci(0) = a (Cf. (60)), (ui,Vi) is the only solution 
 except for a constant factor which satisfies (55), (1) for X = Iq, l^i, Z±2, 
 • • • but for no others. (See Professor Hurwitz's article.) These values 
 separate the X-axis into the intervals 
 
 • • • '(l-n, U+l), • • • (/-I, k), (/o, /l), • • • (In, In+l), ' ' ' (84) 
 
 By (62) at X = one of these values, say Ir, 
 
 P12P21 = L (85) 
 
 And by (76) 
 
 D{lr) = ^ (1 - ^2l)^ r = 0, ± 1, ± 2, . . .. (86) 
 
 And the rest of the proof is evident. 
 
 Professor Hurwitz has shown that Pn has no double roots, also that for 
 the system (1) and 
 
 :?:S::: 
 
 P21 has no double roots. Hence Pn, P21 change sign when they vanish. 
 Again since 
 
 P11P2IA ~ PlU^2l = Wl»lA ~ ^iWiA^x-l. 
 
 by (67) this cannot vanish but is negative. From these facts it is easy to 
 show that the roots of Pn, P21 separate each other as well as do those of 
 
44 Camp: An Extension of the SturTn-IAouville Expansion. 
 
 Pn\, -Pau- Clearly then P21 alternates in sign at the values • • • I-2, Z_i, lo, 
 l\, h, • • ' and we have two cases : 
 
 Case I: | ^^^^ < q at X = /o, /.2, • • • } ^'^'^ ^" > for X = /:; 
 
 Case II: \ 7^., . -^ n x \ ?* ' 7^ ' f when P21 < for X = h. 
 
 \ lf{\) = U at A = /o, 1^2, • • ■ J 
 
 There must obviously exist roots of Z)(X) as follows: 
 
 In Case I — at least two values X„, X„+i in each double interval 
 
 ihp, I2P+2), p = 0, d= 1, =t 2, • • •, 
 
 such that 
 
 hp <C X„ = hp+i = Xti+i <C hp+2 (88) 
 
 and at least one such that 
 
 h^\<l2. 
 
 In Case II — at least two values X^, X^+i in each double interval 
 
 (hp-i, hp+i), p = 0, ± 1, ± 2, • • •, 
 
 such that 
 
 hp—l K \n = hp "^ \n+l < hp+l- (89) 
 
 To show that there are exactly two roots in (kp, I2P+2) we make use of the 
 separative property of the roots of Pn, P21 and their derivatives. Obviously 
 there must be an even number of roots, since we count a double root as two. 
 If a double root occurs it must fall at kp+i bj^ (64) and Lemma III. Then 
 by Lemmas II and I there can be no root elsewhere. If there is no double 
 root, there must be less than four roots, for otherwise there would be at 
 least two in one of the intervals (hp, hp+i), (hp+i, hp+2), which violates 
 Lemma I. 
 
 Similarly we can show that there is exactly one root in the interval 
 (/i, h) and exactly two for Case II in the intervals (hp-i, hp+i) provided we 
 count a double root at h once each in the intervals (h, h), (h, h)- Thus the 
 Theorem is proved. 
 
 Corollary. For |X| large, 0(1/X„) = 0{\lmr), \n\ large. 
 
 Professor Hurwitz has shown that for |X| large. In = nir -\- di -\- 0{ljn), 
 where di is a constant. Thus the intervals (/„, 4+2) for | X | , \n\ large are 
 of length 27r to within 0{\Jn). In each of these intervals for n odd or even 
 according to Case I or II by the Theorem there exist just two values X^, 
 Xm+i of X which are roots of D(\). Since in any finite interval of the X-axis 
 there are but a finite number of roots Ir or X,„, for | n \ large enough we have 
 
 X = ln+8 — k, ^ \k\ ^ T, 
 
Camp: An Extension of the Sturm-Liouville Expansion. 45 
 
 where * is a finite integer positive or negative, provided we begin to count 
 the /'s and X's so that /o = 0, Xo = and /_i < 0, X_i < 0. Clearly then 
 
 X„ = nir + *x - ^ + 01 + a [^\ (90) 
 
 and since mtt is the dominant part, 
 
 Ky^K^J'"-^"- 
 
 Theorem V: // {un, Vn) corresponds to Xn defined as in the premoiis Coroll- 
 ary, then this solution of (1), (2) takes the asymptotic form 
 
 iin{x) = sin \jiTX -\- P{x)'] + ( - |, 
 
 Vnix) = COS {jlTTX + P(x)] + ^ ( " ). 
 
 (91) 
 
 when \n\ is large. 
 
 By the Corollary of Theorem III D(K) will have a root when 
 
 or when 
 
 - C 
 
 (92) 
 
 ^(1) + ^ = sin-»-p=^+ O^iJ, (93) 
 
 since by Lemma V, Section I, \C\ ^ '^A} + B^, and by Theorem IV the 
 right member of (92) must be numerically ^ 1, also. 
 
 and for X = X„, 
 
 »a)-"(i)-»(-;)- 
 
 Hence from (93), since by (58) C > 0, Z)(X) = when 
 and for vi odd 
 
 Xn+i = (m + l)7r - i/' - fn^ — dx— (p-\- 0[-\, 
 
 in which ^ is the angle in radian measure between and 7r/2 inclusive 
 
46 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 whose sine is | C | / -ylAF-^-B^. Clearly (94) may be expressed as 
 
 \n=mT-\-(- 1)-+V- fo'"^ dx- ^+0(y\,^l\n, (95) 
 where \n\, \Xn.\ are large and m = n -{- s. From (42) 
 u(x) = sm(^-eo) + o(^Y 
 
 '0{x) = cos {^ - do) + (^Y 
 
 provided we choose 
 
 a = — sin ^0, ^ = cos ^o- 
 
 This is always possible if we divide w(0), v{0) by ^[^(0)]]^ + C2'(0)]]^ and 
 call the quotients a, ^ respectively. Then by (95) 
 
 Unix) = sin \jmrx + 62X + ^fo^'Z^^is) + his)^^ — Bo'] + I -j , 
 
 Vn{x) = cos [m,irx + d2X + ^JV[_a{s) + b{s)2ds — 5o] + ^ ( - J, 
 where 
 
 62 ^ (- 1)"*+^ - fo' ^ dx- <p 
 
 and the formulae (91) follow, provided we define 
 
 P(x) ^ d2X + STX + fo "^^^ "^ ^^'^ ds - do, 
 
 since m = ti + *, ^2 = — ^- + 0i as shown bj^ (90) . 
 
 Corollary: If we normalize (un, Vn), it will have the same form (91). 
 
 Section IV. Expansion Theorems. 
 Theorem VI: If two otherwise arbitrary functions f(x), g(x) satisfy (2) 
 and have continuous second derivatives, ^ .r ^ 1, then they are expansible 
 in the form 
 
 + 00 
 fix) = ^CnUnix), 
 
 ":r (96) 
 
 gix) = ^CnVnix), 
 n= — 00 
 
 where 
 
 Cn = y*oT/(^)Wn(^) + gix)Vnix)~\dx, (97) 
 
 and {Un, Vn) represents the set of normal orthogonal solutions of (1), (2). 
 
 In case \k is a double value we count it as two and take any two linearly 
 independent normal orthogonal solutions for that value of X. 
 
Camp: An Extension of the Siurm-LiouviUe Expansion. 47 
 
 Let us first show that two solutions {um, Vm), {Un, Vn) of (1), (2) for 
 diflferent values of X are orthogonal, i.e., 
 
 f,'[Um{s)u^{s) + Vm{s)Vn{s)yS = 0. (98) 
 
 It is only necessary to apply Lemmas I, III, Section I, and divide by 
 X^ — X„. Obviously they are easily normalized by dividing in the case of 
 
 (Mm, O by <flJul^^vl)dx. 
 
 If we assume the series of (96) to be uniformly convergent, then 
 
 yoT/(^)*^(-^) + g{x)Vm{x)yix = Cm 
 
 and the series become 
 
 S ^<n(a')yo^[/(^)«7,(ar) + g{x)vn{x)'yix, 
 
 + 00 
 
 S 1^n{x)Jl^[J'(x)Un(x) + g(x)Vn(x)Jdx. 
 
 (99) 
 
 To study these we assume |X„| > |a(a') |, also > J6(a;)|. We have 
 
 y'nix) — (X„ 4- a)Vn{x) = 0, 
 V'nix) + (Xn + b)Ur,{x) = 0. 
 
 Then 
 
 f(x) . 
 So^j{x)Un{x)dx = — Jo^ \ 4- b ^'•(^)^^. 
 
 r MvMT , ^, , ,d[_j(x)_i 
 
 . =L"XTtJ„ + ^«^^"(^)5^[x;:tk^)J^^ 
 
 r fvn T , ^. K(x) dv f -] 
 
 L X„+6jo"^*^° K + adxlK + br''' 
 
 But since by the Corollary of Theorem IV ( — j = ( — ) = (-\ 
 
 for \n\ large, we have 
 
 d ( fix) \ fix) f(x)b'(x) 
 dx\K + b{x )) ^ K+b iK + bJ ^ ^ /J_\ 
 \n + a{x) \n+a \n- J 
 
 Similarly 
 Hence 
 
 '"^ ^ [ x;h=^ ~ rf^ Jo + ^ (^0 
 
48 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 or 
 
 c„ = ^ Igun - fVnJ +0(^\^' (100) 
 
 Since by hypothesis /, g satisfy 
 
 UiU, g) = u,u, g) = 0, (loi) 
 
 by Lemma III, Section I, Cn = 0{l/n^) and each series in (99) converges 
 uniformly. If f{x), g{x) are expansible, we get the series (99) for them. 
 To show that these series converge uniformly to f{x), g{x) respectively, 
 we define 
 
 + 00 
 
 F{x) = Z) Un{x) fQ^[^f{x)Un{x) + g{x)Vn{x)yix, 
 
 n——QO 
 
 Gix) = Zl 'Vn(x)fQ^[_f(x)Unix) + g(x)Vn{x)2dx. 
 
 n= — 00 
 
 Then formally we have 
 
 Jo^[_F{x)Unix).+ G(x)Vn{x)'}dx = Jh^[_f{x)Un{x) + g{x)Vn(x)']dx 
 
 or 
 
 JhHZFix) -f{x)2un{x) + iGix) - g{x)-]vn{x)]dx = 0, 
 
 w = 0, ± 1,±2, .-.. 
 If now we define 
 
 F(x) -fix) ^ rPix), . Gix) - gix) = - ^(x), 
 
 then by Theorem II we have 
 
 Fix) ^ fix), Gix) ^ gix), q.e.d. 
 
 Theorem VII: If fix), gix) do not satisfy (2) hut possess continuous 
 second derivatives as before, the series 
 
 fcoMo(a:) + [ciUiix) + c_iW_i(.t)] + [c^u^ix) -f c_2W_2(a:)] + • • •, qq2) 
 \cQVQix) + [ciViix) + c_i«_i(a:)] + [cic^ix) + c-^v-^ix)'} + • • •, 
 
 converge uniformly to fix), gix) forO<€^a:^l — e<l, where e is an 
 arbitrarily small positive number and iun, Vn), Cn are defined as before. 
 We have from (100) 
 
 CnUnix) = -y— \igix)Unix) - fix)Vnix)Jii -\- \—A ' 
 
 Hence by (91) for ln| large 
 
 , . (- l)''KiUnix) - KoUnix) , 
 CnUnix) = — h 
 
 "(i) 
 
 where 
 
 Xi^5r(l)sinP(l)-/(l)cosP(l), 
 Ko = giO) sin P(0) - /(O) cos P(0). 
 
But 
 
 Camp: An Extension of the Sturm-Liouville Expansion. 49 
 
 Again 
 
 ,, (- l^Kiu^jx) - Kou^jx) . ^/ 1 \ 
 
 c^u.M = zr^ + ^ U j • 
 
 Hence 
 
 Cn^niir) + C-nU-n(x) = ^ ^ J •-"" ~ """-J" 
 
 ?/„(.r) — u-„(x) = sin [nxa- + P(a-)] — sin [— wttx + P(a-)] + ^M ~ ) 
 = 2 sin (nirx) cos P(a-) + ^^ ( " ) ' 
 
 •'• r„u„(x) + C^U-„{x) 
 
 T^, s r rr / ,s sin nxa: _, sinnxj;~| , ,. / 1 \ 
 = 2 cos PW [ A,(- D- -^^ - A. ^^ J + (;^) • 
 
 ^, • v> (~ 1)* sin nTTj; -^ sin nirx .. . , 
 
 Ihe series 2^ ' Z^ converge uniformly, for 
 
 ,; = l nw ^x 71T 
 
 0<€^x^l — €< 1, and since we get a similar expression for 
 
 the rest of the proof is like in the Theorem above. 
 
 Section V. Further Results on the Distribution of Principal Parameter 
 
 Values. 
 Witliout the use of Theorem IV we can show that for the case in which 
 
 \C\ < ^IA^+ B\ 
 
 D{\) has exactly two roots in each interval 
 
 Q2A- — l)7r, (2k + l)7r], \k\ large and integral, 
 
 which have the asymptotic form (95) for \n\, |X| sufficiently large. With- 
 out loss of generality assume (58), then by definition C = 2, Then 
 
 ^1A^+~B' -2 = E>0. (103) 
 
 By (50) 
 
 D{\) = Vl2+^sin [^(1) + ^] 4- C + ^' ' 0^\K\ < M, 
 
 for 1 X 1 large, or 
 
 D{\) ... C _,K' 
 = sin A + • , + -T- ' 
 
 VlH^' ^A' + B' ^ 
 
 where 
 
 A = ^(1) + <p. (104) 
 
 For A = (2A- — l)7r or 2A-ir, and [Xj > Xj/, 
 
50 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 For A = (4:k - l)7r/2, and |X| > X^, 
 
 ^(^) =-i+-^i^+g-'= -^ +g:<o. 
 
 Let Xi be the larger of \m, X^, then for |X| < Li, D(\) must have at 
 least two roots in the intervals (21- — l)7r < X < 2kT. Since for 2kw ^ A 
 ^ (2^ + l)7r, sin A ^ 0, D{\) has no root there. To show that there are 
 exactly two roots in the previous interval we use the fact that Z)(X) = 
 only when 
 
 sin A = — , + — • 
 
 V^2 _^ ^2 X 
 
 It is obvious that for | X | > some L2 the right member is always negative 
 since \K'\ < M. If L is the larger of Li, L2, then, since 
 
 sm A = — 1 + — ' 
 
 ^ji2 _^ ^2 X 
 
 and when | X | > Xjs; 
 
 K^ E ^ 
 
 ^ V^2 _^ 52 ' 
 
 we have 
 
 > sinA > - 1, |X| > i. 
 
 Obviously there are just two values of A in any [(2^• — l)7r, 2A-7r] 
 interval which satisfy this inequality, i.e., 
 
 Ai = {2k- i)t + iy -^ (^y 
 
 A2 = 2kTr-iP+ of y\, 0<iA<v 
 where sin j^ = 
 
 .'. A = W7r+ (- 1)"»+V+ o/'i'j, 
 for \m\ large, and by the definition of A 
 
 Xn= m7r+ (- l)m+i^- f.^^^^dx- cp+0(-), 
 
 2 \fnj 
 
 when 1^1 is large; or since in each 27r-interval there are 2 values of X 
 we have 
 
 Xn= n7r + 57r+ (- l)"+«+V - fo'^^ dx - <p + (-), 
 
 s being a finite integer. 
 
 Case of C^ = A^ -\- B^: Here the above reasoning does not apply and 
 we attack the problem by getting a more accurate asymptotic expansion 
 
Camp: An Extension of the Sturm-Liouville Expansion. 
 for D{\) and D'(\). If we assume 
 
 51 
 
 v(x) = e^ 
 
 '" + X + V + 
 
 J 
 
 then substituting in (1) equate coefficients, we are led to the expansion 
 
 i(x) = Aoe^* + 5oe-«* + Z) 
 
 A^e^'+B^-^' 
 
 t{x) = W + Doe-^' + Z 
 
 A^l X* 
 
 or since we restrict X to real values, to 
 
 f \ — A y I T) • ^ , \^ Ak cos ^ + Bksin ^ 
 
 u{x) = ^0 cos ^ + JSo sm ^ + 2w \t • ' 
 
 *=i X* 
 
 r \ — ri >■ I n • v , v^ C** COS ^ + D* sin ^ 
 x{x) = Co cos ^ + Do sin ^ + 2^ ^— ^ ^ > 
 
 where ^ is defined as in (44) and A, B, C, D are independent of X. 
 We may show that (ay) = (/35), (a8) = (y^) so that 
 
 yl = 2(a7), ^5 = 2(a5), C = 2(«/3), 
 
 and since (a/9) = 1, we have 
 
 D{\) 
 
 (105) 
 
 V^2 _|_ ^2 
 
 = 1 + [2^(1) — «i(l)]| cos v? + Cwi(l) + «2(l)Ili sin <p, 
 
 in which (wi, vi), (1^2, ^2) are defined by (12) and <p by (52). F'rom (50) 
 we can show that D'{X) has a root in the interval 
 
 (2k - l)7r + iTT < A < {2k - l)7r + It. 
 
 (106) 
 
 By putting (105) in (1) and equating coefficients we get a set of equations 
 from which we can solve for A, B, C, D and if we put that value of X in 
 for which Z)'(X) = 0, we obtain 
 
 D{\)- 
 
 m\ + 2mQm\ cos 2ip -\- ml 
 
 8X2 
 
 wherein 
 
 Hi) 
 
 mi = \a{\) - \h{\), wio = |a(0) - ^6(0). 
 
 Hence Z)(X) will have two roots in the interval (106) unless 
 
 Case 1) Wo = wi = 0; 
 
 Case 2) (p = cos~^ {ay) = sin~^ (a5) = It, diq = mi; 
 
 Case 3) <p = 0, mo = — mi. 
 
52 Camp: An Extension of the Sturm-Liouville Expansion. 
 
 By extending the calculation to the fifth set of functions Ai, Bi, C4, Di 
 we find that D(X) will still have two roots in (106) for Case 1) unless 
 
 nil + 2momi cos 2^ + m^ = 0, 
 i.e., unless 
 
 .1°. m'o = m[ = 0; 
 
 2°. cp = Itt, mo = m[; 
 
 3°. v' = 0, m,) = — mi; 
 in which 
 
 m', ^ ia'(O) - i6'(0), mi ^ ia'(l) - i6'(l). 
 
 The same is true for Case 2) unless 
 
 (m; - m',y + 4m^(/i - kY = 
 or 
 
 1°. m'l = mo, mo = 0; 
 2°. m'l = mo, /i = lo; 
 where 
 
 /i^|a(l) + P(l), /o=|a(0)+|6(0). 
 
 It is probable that by this method D{\) would also be shown to have 
 roots for Case 3) unless 
 
 (mi + m,y + 4ml{h + loY = 0, 
 
 and that if one should continue the computation one would find that D(K) 
 possesses roots unless 
 
 I. mo = mi = m| = m,, = mj' = m^' • • • =0; 
 II. (p = ^TT, mo = mi, m[, = m[, m[,' = ml', • • • ; 
 III. (p = 0, mo = — mi, ttIq = — m[, tyiq = — m,", • • •. 
 
 It is interesting to note that for a restricted case of I, namely 
 
 a{x) ^ b{x) ^ 0, 
 
 D(X) possesses only double roots, provided 
 
 (107) 
 
 ai = Ti = 1^2 = §2 = 1, 
 q;2 = 72 = /3i = 5i = 0, 
 
 Here 
 
 2}(X) = 2 + 2 cos X, 
 
 X = (2^ - l)7r, 
 in fact 
 
 ^ 1 + cos X, sin X 
 
 — sin X, 1 + cos X 
 
 This leads to a special case of Fourier's Series in which the terms involving 
 sin mrx, cos nirx for n even are wanting. 
 
Camp: An Extension of the Sturm-Liouville Expansion. 
 
 53 
 
 Another interesting result is the following 
 Theorem : A necessary condition that the determinant 
 
 D(\)^ 
 
 
 he of rank zero is that (a/3) = (78), where (wi, v\), (ih, V2) are defined by (12). 
 We have 
 
 UiiuiVi) = ai-\- 7i?/i(l) + 5i»i(l) = 0, 
 
 U2(uiVi) = <X2-{- y2th{l) + MKl) = 0, 
 Ul{U2V2) = fii + TiW2(l) + 8MI) = 0, 
 
 U2(U2V2) = ^2+ 72«2(1) + 52»2(1) = 0, 
 
 Multiply the first two equations by 82, — 81 respectively, then 
 (ad) + (t5)wi(1) = 0. 
 Similarly using — a2, «i on the last two we have 
 
 (a/S) + (ay)u2a) + (a8)v2{l) = 0. 
 Again using 72, — 71 to multiply the first two, we get 
 
 (ay) - (y8)vi(l) = 0. 
 Combini'ng equations (112), (114), (113) we obtain 
 
 (108) 
 (109) 
 (110) 
 (111) 
 
 (112) 
 
 (113) 
 
 (114) 
 
 Mi(l), vx(l) 
 
 1*2(1), ^2(1) 
 
 or 
 
 (a^) - (yd) 
 
 (afi) = (y8). 
 
 = 0, 
 
 Q.e.d. 
 
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