LINU.

UNCLASSIFIED
ORNL
.



:
:

310
IN
UN
OR Nh-p-310
SEP 2 i 1964
MASTER
A STUDY OF SOLUTIONS OF BENSSELSLATION
A Tesis
Presented to
the Graduate Council of
The University of Tennessee
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
by
Joseph Lewis Hosszu
August 1964
July 17, 1964
To the Graduate Council:
I am submitting herewith a thesis written by Joseph Lewis Hosszu
entitled "A Study of Solutions of Bessel's Equation." I recommend that
1t be accepted for nine quarter hours of credit in partial fulfillment
of the requirements for the degree of Master of Science, with a major in
Physics

W.E. Deeda
Major Professor
We have read this thesis and
recommend its acceptance:
Accepted for the Council:
Dean of the Graduate School
ACKNOWLEDGEMENTS
The author wishes to express his thanks to Dr. W. E. Deeds for
suggesting the topic and for valuable criticism during the preparation
of this dissertation.
Thanks are also due to Drs. R. J Lovell and W. R
Bugg for their valuable criticism of this dissertation and to Dr. M. L.
Randolpli of the Biology Division at Oak Ridge National laboratory where
much of the computational work was performed.

TABLE OF CONTENTS
CHAPTER
PAGE
I.
Introduction ..........
II. Mathematical Properties of Bessel functions ......
III.
Construction of Demonstration Models ..........
IV.
Calculation of Tables. .................
To to ū un ūdenone
V.
Summary
.
.
.
.
.
.
.
.
.
.
.
.
.
.
VI. Bibliography ......
...............
.
APPENDIX A
.
..........................
.
APPENDIX
B
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
•
.
ili
LIST OF TAPLI
TABLE
PAGL
43
I. Expansion of the Integral Order Besccl Functions .....
II. Lxpansion of the Negative Half-Integral Order Bessel
Functions........................
44
III. Expansion of the positive Half-Integral Order Brasel
Functions........................
45
IV. Expansion of the Negative Integral Order Spherical
Bessel Functions ....................
46
V. Expansion of the Positive Integral Order Spherical
Bessel Functions . . . .
.
.
.
.
.
.
.
.
.
VI. Bessel Functions of Positive Integral Order. .....
VII Sessel Functions of Negative Integral Order. ....
VIII. Bessel Functions of Negative Half-Integral Order ...
IX. Bessel Functions of Positive Half-Integral Order .....
X. Spherical Bessel Functions of Positive Integral Order...
XI. Spherical Bessel Functions of Negative Integral Order. ..
XII. Normalized Bessel Functions of Order Zero. ........
iv

CHAPTER I
INTRODUCTION
In the study of mathematical physics, or in any field of science
in which mathematical expressions must be applied to a set of physical
conditions, it is generally desirable to obtain a general formulation
or equation into which an array of parameters may be inseried to fit a
specific case.
In this manner a student learns to associate the basic
principles which are comoon to what superficially appear to be highly
diverse phenomeng.
Since the criteria by which a particular theorem 18 judged are
ultimately based on experimental evidence, the applications of mathe-
matical models to the real world 18 one of the primary concerns of the
mathematical physicist. At times the theory precedes experimental
verification and at other times it lags behind. Whether the theoriat
obtains an equation derived from first principles or empirically deter-
mines & formulation which organizes a body of data, his application of
experimental data obtained under one set of physical conditions to a
mathematical model which was derived to explain a quite different set
of conditions often leads to further insight into the world around us.
One of the most widely used methods of such syntheses is based
on the geometry of the system. It is quite often convenient to con-
sider as a class those problems which are spherically or cylindrically
symmetric. This method does have certain inherent difficulties, but
with due allowance for them, It is & rather powerful tool for many
purposes.
The description of phenomena in terms of the interaction between
the variables of a system can generally be best written as a differ-
ential equation.
The difficulty of solution of sucli a differential
equation depends to a considerable extent upon the number of the system
parameters
An adequate understanding of the basic mathematical principles
and operations is therefore a basic key to the understanding or descrip-
tion of a physical condition or phenomenon.
It is difficult to realize that today an average student cen use
and solve differential equations which were considered to be very diffi-
cult of solution even by the best rathematicians a few generations ago.
Part of the reason, of course, is that those mathematicians did solve
the equations, and the methods which they used are in many cases still
used. Also, those mathematicians were influenced by the then current
philosophy of thought and tried to express their solutions in terms of
finite quantities. The use of infinite series and transcendental
functions to express the solution to a problem is a relatively recent
innovation.
One of the more difficult or these differential equations was
one of the first order, known as Riccati's equation. This equation,
dy
= A + By + cya,
dx
does not yield a solution which is expressible in finite terms. A vari-
ation on this equation, achieved by an elementary transformation, is a
linear differential equation of the second order known as Bessel's
equation. A Bessel function 18 the term applied to a particular
solution of the Bessel equation.
This solution of a single equation, originally derived to serve
as a general expression or description of the path of the radius vector
in planetary motion, 18 an excellent example of the wide range of appli-
cability of a single type of equation in fields of study which are widely
divorced from the original area of investigation.
Watson (5) cites as introductory material to the treatise on
Bessel functions the example of Daniel Bernoulli who in 1732 succeeded
in deriving a general equation to describe the oscillations of a heavy
flexible chain, L. Euler who in 1764 derived the general expression for
the vibration of a stretched membrane, J. L. de Lagrange who in 1770
described a derivation for the orbital motion of the planets, and F. W.
Pourier who in .1822 derived an expression for the cooling of a hot
cylinder.
These phenomena seem to be completely unrelated to each other
and to the equation derived in 1824 by F. W. Bessel regarding the
elliptical motion of the planets.
It was shown, however, in the now famous memoir written in 1824
and published in 1826 by F. W. Bessel that these physical problems were
in fact particular solutions to a general differential equation. The
differential equation was that known as Riccati's equation. Bessel,
however, realized the necessity of using an infinite series solution
with adjustable parameters, or boundary conditions.
Another application of this equation and its solutions is to the
study of the electrostatic potential and field of a cylindrical con-
ductor. Understanding of this particular application is often limited
by the inability to visualize the form of the potential functions in
terms of the Bessel functions. Because of long familiarity with the
elementary trigonometric functions such as the sine, cosine, and
tangent, most students can readily visualize their form either alone or
in combination. Such a familiarity with the Bt8sel functions would
also be advantageous.
Unfortunately, most mathematical texts are not adequate for this
purpose. Several excellent treatises on Bessel functions are available,
but they are usually so comprehensive that the basic structure of the --
functions and their properties are obscured. Otherwise excellent texts
used in mathematical physics are unusable because, in the attempts to
present a representative collection of theorems and methods of solution,
much of the significant detail and explanation is omitted.
This paper is directed at a position somewhere between these
extremes. The solution of the basic equations, some of the elementary
transformations and properties of the Bessel functions, expansions in
terms of a power series, some numerical tabulations, and models or the
functions are presented in a form which the student can grasp and with
enough detail so that the student can easily see the application to
other types of problems.
CHAPTER II
MATHEMATICAL PROPERTIES OP 239 L FUNCTIONS
One of the most widely used equations in physics 18 that bomo
as Laplace's equation. Using operator notation, in which is the
gradient taken with respect to spatial coordinates and ya is the di-
vergence of the gradient, Laplace's equation is written as
in which
16 some function of the coordinates upon which time cgerator
acts
It is customery, in rectangular Cartesiani coordinates, to express
♡ = (x,y,z) and to write eg. (1) as
pº(693) -
We may apply a transformation of the form
x = r cos o
y = r sin
2 =
2
where r and 8 are real.
Upon substitution of eg. (3) into eq. (2), it follows that
D
(1,0,z)
-
"=0.
(..)
r مره
همه م r
چه
For ease in solution, it le generally assumed that (1,?,2) is
a separable product function of the form
(5)
(8,0, ) = R(r)(A)2(2).
Rewriting eq. (4) after substituting the product function R(r)-(0)26-)
for (r,0,2) from eq. (5) and dividing through by the product function,
Laplace's equation becomes
T
OR
JA
g
1
.
.
+
-
o
0
.
r فرن R
ومر r
هو
قيل :
But since each factor of the product function R(r) - R (0) = ,
2/2) = 2y la independent nf the other factors, the partial derivatives
can be replaced by the total derivatives with no 1088 of generality,
and eq. (6) becomes
1. PR
1 dR
1 da z
(8)
R`ar rar po doo z az
Taking the last term of the left-hand-side of eq. (7) to the
right-hand-side allows eq. (7) to be written as
1,Ride
1 daz
R`dr² rar' pa do 2 az
Since the left-hand-side of eq. (8) 18 independent of 2(2), the right-
hand-side must also be independent of 2(2). It further follows that
since the right-hand-side of cq. (8) is independent of R(r) and ele),
the left-hand-side must also be independent of R(r) and m(o). This
maitwal independence of the two sides of eq. (8) requires that each side
ve equal to some constant, say A. .
Thus it iollows that
I da 2
= A
2 diz
and upon rewriting eg. (8)
1, dar la
( + - ) +
Rar rdr
1
den
=
.
A.
radga
After multiplying each side of eq. (10) by read and adding Ard and
-(1/2)dam/age to each side, we find that
je, dR 1 an
-6 +=) + Ane ..--.
- R aro r dr
(11)
A doo
Again applying the reasoning used above, since the left-hand-
side of eq. (11) 18 independent of Ale) the right-hand- side must also
be independent of Fle). Also, since the right-hand-side of eq. (11) 18
independent of R(r), the left-hand-side must also be independent of R(r).
This mitual independence of the two sides of eq. (11) requires that each
side of the equation be equal to some constant, say B.
Thus it follows that
1 de
-
das
B,
(12)
and
1
1. do
-(
Rºdra
1 de
+- ) +
r dr'
B + A
= 0,
(13)
pad
or
med
de R dR
tr - + (B + Ara) R = 0.
dre dr
(14)
If now the constants A and B are chosen such that A = 12 and
B = -x, where da and na may be either positive or negative, real or
imaginary, or even complex, then eq. (9) and (10) may be written
respectively as
qaz
(15)
a la z
ميه
and
do
u-n
.
(16)
dorit
The eq. (15) and (16) can be shown to have the solutions
2(z) - ce iz + De-12
(27)
and
(18)
me) = E cos n o + P sin ne,
where C, D, E, and F are constants which are to be determined from
the initial or boundary conditions of a particular problem.
Equation (14) now becomes
[ ᏛᏒ aᏒ .
pa +r + (- m ) R = 0.
- dr dr
(19)
Upon setting x= nr and assuming that x and n are each real, it can
be written as
de R dR
to come + x
- ax ax
+ (2a - m )R = 0.
(20)
Equation (20) 18 generally known as Bessel's differential
equation and the solutions to eq. (20) are called the cylinder functions
or the Beesel functions of order n.
Ir R () is the Bessel function of order n, the functions
Bin
- Ry (hr){ van (ne) }e tuz
(21)
COS
are solutions of Laplace's equation and are known as a cylindrical
harmonics. For n = 0, the harmonic is symmetric about the z-axis, as
is apparent from inspection of eq. (21).
To solve Bessel's equation, let us rewrite eq. (20) with
DR
da R
Then
x@y" + xy' + (x - poly = 0.
An obvious trial solution of eq. (23) 18 a series of the form
30
Substitution of eq. (24) into eq. (23) ylelds, on the left-hand-side,
the expression
sol E 2 (8 + x)(p + = 1)x1 + x
2 (P + r)x*
r
*
O
+ (x@ - na )xP
8 hill
A
or, after some rearrangement of terms,
- na
Pag (p® - 50) + xD + 1 % (8 + 2)a - mo?
+ x (0:1 (p + r)* - ry] - -
(25)
r
=
2
In order that the eq. (24) should satisfy eq. (23), it is neces-
sary that the expression (25) be identically equal to zero. For the
non-trivial cases this will be true with x 0, only if the coefficients
of all powers of x in the expression (25) are themselves identically
10
zero.
Thus a, (p3 - mo) = 0. But & is the first coefficient in the
series of eq. (24) and 18 itself therefore not equal to zero. There-
.
.
fore
po emo o
(26)
This 18 known as the indicial equation as its solution yields the values
of p, which are of course t n.
Now, taking the coefficients of xp *t, xo*?, ... in the
expression (25) and setting them individually equal to zero yields
ax[(P + 1)2 - ] -0
af (p + 2) op] +-0
(278)
af (P + rjo - be] + 2-2-0,
2 (P + n + 1)(p - n + 1) = 0
Dung (P + n + 2)(p - n + 2) - -
(270)
a(p + n + r)(p - n + r) = -
-2
from which the coefficients by, Bugs Bung • • can be found in terms of
By: From the first equation there results ay = 0, and ag, Big, am and
all subsequent coefficients of odd index are equal to zero. From the
.
other equations, It follows that
ay "
Ton + 2)(p - n + 2)
ao
a
+ more
4
rip + n + 2)(p - n + 2)][ (F + n + 4) (p - n + 4)]
(28)
(-1)*ao
2r
Clp + n + 2)(p + n + 4) ...(p+ n + 2r)]
[(P + n + 2)(p - n + 4). . . (pon - 2r)]
• The coefficients, in terms of ag of eq. (28) satisfy eq. (27)
and the right-hand-side of eq. (24) reduces to
xaldo - no).
Hence eg. (23) satisfies the relation
| xy + xy + (xe - nê ly - xe ( - ) . (29)
Now, if in eq. (27) and (24) one allows p = I n, the series
solution of eq. (24) will be a solution of eq. (29) or eq. (23), but
only if the series of eq. (24) 18 convergent.
If p = + n, then
ya agen 2. anta) * zuulen + amlən ...}. (50)
y
: 2,X"
1-
. (30
2(n + 2)
2.4(2n + 2)(2n + 4)
There can be found a similar solution by replacing p = +n with p = -n
in the series, from the other root of the indicial equation, and it
will be distinct, provided that n is not an integer.
Applying the ratio test for convergence, one can write the
recurrence relationship of the coefficients as
Prºr(2n + r)
Starting with a = 1 as roc
-
O,
roxas
r(2n + r)
Bo that tñe series converges for all values of x.
One usually sets a = 1/2"(n.) 18 n 18 a positive integer, or
Ao = 1/2" r(n + 1) 18 n 18 unrestricted. For the former case denoting
the function as Jo (x):
28.1!(n + 1) 24.2!(n + 1)(n + 2)
(32)
it...}.
20.3!in + 1)(n + 2)(n + 3)
It 18 apparent that Jn (x) 18 generalized for all values of n except
the negative integers by writing r(n + 1.) for n! .
Then we would have ag - 1/2" r(n + 1) and
co (-1)" yn + 2r
Jn(x) = E .
roor!r(r. + r + 1)
If n is not an integer, p = -n yields
2r - n
(g(x) =
(-1)"_
r = 0 rir(r - n + 1)
2
Thus, if n is not an integer, the general solution of the Bessel's
.
.
equation is
y G Jn (x) + H J-n(x) ·
where G and H are arbitrarily chosen constants.
23
However, if the value of n 18 either zero or an integer, the
solutions of eq. (33) and (34) are not distinct. If n 18 zero they are
obviously identical. If n 18 a positive integer, the first n coeffi-
cients of eq. (26) vanish since 1/1(- p) 18 zero for p a positive Inte-
ger and eq. (33) becomes
(-1)
- n + 2r
(x) = 0 + f
(-) .
ranrir(r = n + 1)
or, 18 ron +8:
n
+ 28
__$(x) = (- 2)". 2. 3) ***,
8 = 0 (n + 8)!!!
from which it can be seen that:
5-(x) = (- 1)" J (x) ·
(36)
The Bessel function solutions are seen to depend upon a recur-
rence relation between the coefficients of a trial series solution. It
would be convenient 1f the Bessel function solutions of the first kind,
Jo (x), were also bound by some form of recurrence relationship. Fortu-
nately this is the case. With y' = dy/ax, eq. (23) can be written as
vity (1
- .
Now, let y = x". Then, with v' = dv/dx,
2n + 1
' v = 0.
To find v, let us assume that there exists a series solution of
the form
11
vs £ box
(39)
and substitute eq. (39) into eq. (38). Then
i [m(2n + m)bux" - 2 + bukty - O.
(40)
m
= 0
Since a non-trivial solution 18 des Ired, it is necessary that
Om - 2 = -1(2n + m) bm. If we choose to begin with m = 0, then bm- 2 =
O, BM-4*0,..., and
2
2(2n + 2)
22 (n + 1)
DO
ST
+ 2.4(2n + 2)(2n + 4) 2!2 (n + 1)(n + 2)
This just duplicates the coefficients of the Bessel function series
solution, as derived earlier. It 18 easily seen that xv 18 a solution
of the Bessel equation for any n.
It can be shown that a generating function of the form
XV
<(x/2)u - 1/u).
ve Ov! '2
HM8
atlē
(41)
will yield, after some manipulation, the equation
+ m.
€(x/2)(u - 1/u) = {
"J(x) ·
(12)
na co
..
:
Then substituting for Jn (x) from eq. (33), differentiating and
clearing terms, we find that
x"J}(x) + nx" - 2 Jy(x) = x" In - (x)
(43)
where
() (54(x)],
also,
x"""}(x) = nx- 1 Jy(x) = - **" Jn + z(x),
(44)
These may be written as
5:02) -262) - ICN),
and
J'(x) = - Jn + 2(x) + - Jn(x) ·
X
n
Thus it follows that
2n
29 J/(x) = In - 2(x) + Jn + 2(x).
(45)
Therefore, knowing the value of Jo(x) and J,(x), or any pair of
Ju (x) and Jn + 2(x), one may determine the value of any of the Bessel's
functions whose order 18 a positive integer. Further, by the use of
eq. (36) and (45), one may determine the value of any Bessel function
of any integral order. It can be shown that the Bessel functions can
be written for orders n = 2, 3, 4, 5, . . • in terms of the first ard
zeroth orders. An expansion of these Bessel functions is listed in
Appendix A, Table 1.
With the expressions for the Bessel functions of the various
orders, the values of the expansions are not too difficult to calculate
for the various values of x.
There are several excellent tabulations
of such determinations to many decimal places. These tabulations are
16
expressed as decimal fractione and are actually approximations because
the Bessel functions are really infinite series.
from eq. (32) and (45) with n = 0,1,2,3,4,5, ... and x = 0,1,
2,3,4,5, • . . It can be shown that
•1!(1)
.2.12
28.3!•1•2.3
dolu) - -
• 24-
li 2.31.195* ...}
1)(3)(2)-(3) 3.)**...}.
Since the algebraic signs are alternatively positive and negative, and
monotonically diminishing after a certain term for which (x/2)2 + 18
less than (n!), the series can be terminated at any point after that
term with the knowledge that the subsequent terms are in aggregate less
than the last term of the expansion which is retained..
with the aid of eq. (32) and the expansions of Table 1 of
Appendix A, the values of the Bessel functions of the first kind have
been computed for x = 0,1,2,3,..., 20 and n = 0, 41, 42, ..., 110,
and are listed in Tables VI. and VII of Appendix B.
Because of the multiple uses of the half-integral order Bessel
functions of the first kind, as will be shown later, these will be
chosen for examination as examples of the non-integral order solutions
of Bessel's equation.
The solution of Bessel.'s equation for the half-integral order
Bessel function, particularly that one which is in such a form as to
possess expression in finite terms, has been treated by many authors.
One such solution, based on the work of Lommel and chosen for this
example because of its form, is given below.
n (1)" *n (n + r)!
J-(n + 1/2)(x) = le*
r=0 (n - r)! (2x)*
i n (-1)} + n(n + r)!
r = 0 r!(n - r)! (2x)"
for n any positive integer, will yield after the application of soine
basic trigonometric function expansions
3-(n + 1/2)(x) =
nx sn/2
x + - ) E
2 roo
(-2)*(n + 2r):
(2r)!(n - 2r)! (2x)2r
- 811 (x + y ) 51/2(n = 1)
2 ro
(-3)*(n + 2r + 1)!
(2r + 1)(n - 2r -1)! (2x)
Since excellent tabulations of the sine and cosine exist, the
appropriate insertion with values of n and x very quickiy yiclit the
various values of the Jun+ 12)(x).
The solutions of J+(n + 1/2)(x) and J-(n + 1/2)(x) with n a
positive integer, are distinct.
The expansion of the Juin
(x) are listed in Table II of
Appendix A.
By a similar method, it 18 possible to determine the values of
the positive half-integer cylinder Bessel functicns of the first kind.
The general solution for these functions are more conveniently
18
determined 1!. n is restricted to be a positive integer equal to or
greater than zero.
Then
12
(x/2)n + 1/2
+ I Loull
'n + 10(x) =
(48)
xt (1 - ) Mat .
nyx - 1
Or, upon integration by parts (2n + 1) times,
n (1)! -n-(n + r)!
'n + 1/2(x) =-
r=0 r!(n - r)! (2x)
te-ix
(-1)- nol(n + r)? (49)
r=0
r!(n - r.)!(2x)"
which can be rewritten as
sin
'n + 1/2(x) ::
nx
2
s n/2 (- 1)*(n + 2r)!
r = 0 (2r)!(n - 2r):(2x)ar
s 1/2(n-1)
+ COB ( x
-
(50)
- 2 . r . I
(- 1)*(n + 2r + 1)!
(2r + 1)!(n - 2r - 1)?(2x)2 r *
The expansion of eq. (50), as listed in 'Table III of Appendix A,
can then be readily used to express the numerical values of the positive
half-integral order Bessel functions.
These tabulations are to be
found with those of the negative half-integral order values in
Tables VIII and LX of Appendix B.
An isometric drawing of the integral and half-integral order
Bessel functions of the first kind appears in Figure 1.
It was mentioned earlier that the choice of the non-integral
order solution was made because there existed a multiplicity of uses of

.
.
ID:
hubad ONLU
pything in pija MT!
AU
1:1
TIHTI T.
TWIT
T H Toti
.
in
1
"
DU
Turn 11.
-
Lot,
-
-
torin t
orty na, noviny
i Ri
omning
I
-
-
-
.
.
mico-
-
IND
!!!
14 dubbio
Without
Whitrip
.
10
.
IHMIS
Mid
!
hahid
11
. .
inti
-------
. .
.
--
.
L
.
-
-
-
.
_
.
-
-.-
-
1
.
-
-
ht
; f.. . :
:1.,!!!
..
...
..
.
...
... ...
Figure 1. An Isometric Contour Plot of the J.(x).
CV
20
the half-integral order colution. In this connection, let us consider
the following derivation, which, though quite elementary, does show an
important application of the halt-integrul order Bessel functions.
There exist problens, due to geometrical considerations, whose
solutions and form are more easily expressible in terms of spherical
coordinates than in terms of rectangular coordinates. To solve these
problems, the Laplacian lo expressed in terins of spherical coordinates
with the transformation
x = r sin a cos
y = r sin o sin
(51)
2 = r cos 0.
Then the Laplacian 18 written as
1
Die
SPD
amb el Connect
sin
mor. dr
pod sine do
(52)
* Podporou
For convenience of notation, let the following operators be defined:
Pero
12 = --
lo con og la
(sinoe).
den
(33)
(532)
sino
do
sin od 83
and
L = - 12.
(530)
Substitution of cq. (53a) into eq. (52) yields
para el
21
Allowing 0
to operate on , as defined earlier, Laplace's
equation takes the forms:
Divo
0,
It is convenient to assume, as was done earlier, that the
function y(r,0,?) 18 separable into a product function of the form
(,0,) = R(r)y(0,)
(58)
After eq. (58) 10 substituted into eq. (57) and the result is
divided by the product function R(r)Y(0,4) and the result may be
written as
1
2
3
*) =
Y .
(59)
Ror'Or
Y
Since the left-hand-side of eq. (59) is independent or 0 and $,
and since the right-hand-side of eq. (59) is independent of l', cach
side must be independently equal to some constant, say C. The right-
hand-side of eq. (59) can thus be written
+ IP Y = C
1 Y = CY,
and solution of the differential equation requires that
C-242 + 1)
(60)
for most applications to problems possessing spherical symmetry.
Examination of the left-hand-side of eq. (59) shows then that
262) - £12+ 1).
(61)
Ror' or
Since the only variable present, R(r), is independent of 8 and 9, the
partial derivatives can be written as the total derivatives with no
loss of generality.
Thus
2 (20) = 348 + 1)R
(62)
dr
dr
..-
in-..-.
If the solution of eq. (62) in some way involved the cylindrical
Bessel functions, the result could be more easily expressed. Therefore,
.:-
-
.::.
assume that eq. (62) has some solution of the form
-
.
(63)
where a and a are constants and r and y are functions of r. Then
AR
= cara - 2y + arcy'
: = cura + 2y + ara + 2y!
ar
dr
( - ) = a(a + 1)aray + cara + ly' + ala + 2)x4 + Ly'
- + ara + 2y",
where y' = dy/dr and y" = døy/ame. Then eq. (62) can be expressed as
y" Tare + 2 ] + y' [axer@ + 1 + a(Q + 2)2 + 1]
+y [ack & + 1) - eX(L + 1)]* = 0,
(64)
21a
ala + 1) - U2 + 1) ,
yet ys [ 244 +2" ]+v[ ala * 1,24+2) 3.0. (6)
y" + y'
(65)
.
An examination of eq. (65) shows that since the constant a 18
eliminated during the algebraic manipulation, its value may be arbi-
trarily chosen to be unity. Further examination of eq. (65) to deter-
mine the value of the constant a is not so fruitful. However, if y is
to be expressible as some form of the cylindrical Bessel function
solution, it must be, as are the Jn (x), a Sturm-Liouville eigenfunction.
The Bessel functions themselves are Sturm-Liouville eigenfunctions and,
as such, must possess the property of orthogonality
The condition of orthonormality is easily met if the solution
of the separated radial solution of the spherical Laplacian is finite
at the singular point r = 0. Thus for the orthonormality integral
there results
j[,(r)] [rº!!(r)] +P dr = ønn'.
(66)
But eq. (66) can be satisfied only if it is of the form
s podem (r); (r) ]rar = 8n! .
Thus faz a * ? = r or a = - 1/2.
Replacing a by - 1/2 in eq. (65) yields
21- 1/2 + 1), - 1/21- 1/2 + 1) - 14+ 1),
or
1
(6+ 1/2)2
yo [.(et plat jo-o.
y" + - y' + !
V
s
0
(67)
hero
Examination of eq. (67) reveals that a solution in terms of the
cylindrical Bessel function solutions involves terms of the order of
16+ 1/2). Thus
y = A +54 + 1/2)(r) + BJ-46 + 1/2)(r).
(63)
Hence
R = Ar=1/25+66 + 2/2)() + Br=1/25-66+ 1/2/(r).
(69)
But since R remains finite for r = 0, B = 0. And
A
R
=
(70)
PRO+66 + 1/2)(=)
To determine A, the substitution of eq. (70) into eq. (62)
yields the result that the final answer is independent of A and it may
therefore be arbitrarlly chosen to be unity.
To write the spherical Bessel function solution, jelr), in terms
of an integral order, one may extract the half-order [-function value
and write
je(r) = 1/7/2r J4 + 1/2)(r)
(71)
where je(r) 18 the integral order spherical Bessel function solution.
The expansion of the 39(r), based on eq. (71) and the expansion
of the half-integral order cylindrical Bessel function solutions of
Tables II and III of Appendix A, are listed in Tables IV and V of
Appendix A. The computations of the values of the spherical Bessel
function solutions are to be found in Tables X and XI of Appendix B.
An isometric drawing of the integral order spherical Bessel
functions appears in Figure 2.
Families of functions which possess the property
ro,nim
Syn(x)y_(x)dx = { °.
(72)
An' n = m
are said to be orthogonal over the interval asxsb. If An = 1, for all
n, the set of functions is said to be normalized and is known as an
orthonormal set of functions. The equation above, written in a more
general form, 18
po, ni m
| P(x)y,(x)ym(x)dx = {
(73)
,
n
=
m
with the family of functions said to be orthogonal with respect to the
weighting function p(x). It is often the case that (p(x)]?/?(x),
1 = n, m, is defined as the basic function and eq. (73) reduces to
eq. (72).
The property of orthonormality can casily be shown to apply to
minum2,är?
i
Va
26
er:-

1
Kli
Vi
1
.
:
1
II
B
1
.
,
. . .,:, idee,
.
-
+
--
. .
-:..
..---
.
.
... .
.
.
PES
pismenguetooth
ASKI
finanunturi
IC
23
ESITE:
...
.....
.-
,
"
.
-
th
Lii
f
2
-
-
-2
-
.-
-
..........
U
.
wanita...
Lombardo.....www.
adidas
sodomisel.co... haine de concluded.co.ndo. Is.....imeid.
ماليه المسامه
Figure 2. An Isometric Contour Plot of the Spherical Bessel
Functions of Integral Order.
27
the Bessel functions of the flrøt kind. Into eq. (23) allow J. (OC)
and J (Bx) to be substituted for y. The form of the eq. (23) will then
be, for each case:
de Jo (2x) DJ (ex)
-tx- +(@xa - no )j, (ax) = 0,
dxo
dx
DJ (Bx) DJ (Bx)
>> + (8922 - P )J, (Bx) = 0, (75)
- an ux
with the stipulation that n20. After multiplying eg. (74) by
Jn (Bx)/x and eq. (75) by Jo lax)/x, we find that
de Ju(xx)
, dJax (gx - n )
xJ,(8x) – – + J (8x) - “ – + -- -- Je(x)/(8x)
Xin
axa
dx
(76)
and
(lx)
(82
- if)
dJ, (Bx)
Jolox) -
*()
t
+
Jn (Bx)J (cox)
dx
= 0,
(77)
respectively. Subtraction of eq. (77) from eq. (76) yielus
e le y con el caso
= ($9 - ? )xJ, (ccx))/(2x)
(78)
Equation (70) may then be integrated over the li:nits of 0 to x:
(99 - cp)*,(60)_(8x)ax
= x [4,(@x) apr. com)...com) at (par]
(79)
dx
dx
28
Then, let
J'(r) = -
(30)
nir). Vor)
,
dir
60 that
dIn(Bx)
- BJ'(Bx)
(31)
dx
Then eq. (79) can be written
(-op) !*xJy (Cox) Jy(Bx)dx
(82)
=x[w, (Bx)"}(60) - Buy (20x),(8x)].
Now if a and B are rcots of Jn (), that is, ir Jola) = Jo(B) = 0, and
if a + B, eg. (82) reduces to
| x(x) (3x)dx = 0
=
0
.
(83)
For the case of a = B, differentiation of cq. (82) with respect
to B will yield
o
20]* (60) Jy(8x)ax + (8° -20) * 5% (03) J\(8x)dx
(011)
= x Pocksy (Bx).Jy (oux) - Jn (cex)J; (Bx) - BxJ,(vx)J; ()?.
Now let Q = B and divide both sides of eq: (81) by 20, so that
** [1,(-0.7%ax o facem [15/(0) – 1, Cara) () (05)
- 086), (25)J(cx)} ·
X
How If a is a root of Jh(x) we will have
Dx [ 5, (.) ax = 5, (c) Tº
The condition of orthonormally has 165 uther liian to present u
simple method of obtainine the enerica Buscl functions.
Mariicui:
case is the function which can be catancini in FO! 10! Ceri.wh: 1:16
range from 0 to a. Such a function may be a potential or a firli
boundary in electrostatics. Since the function is expressibl, as uri
expansion, let us expand it in a series of Bessel functions
f(x) =
ſ
=
A, Jn (a,x),
i
1
where the ar's arc chosen such that J, la, a) = 0. To deterrine the
value of Ay, it is necessary to multiply both sides or er:. (07) b;
XJ (04x) and integrate from x = 0 10 x = a. Bis the orthogona' it!
property all terms on the right-hand-side vanisii, except A.
x J. (x)] so that
5°xF(x)(4xax
4 =0
(w)
[Jn (@_x)]@chx
For the particular case of the Bessel lincinte, che
vieirii...
of orthonormality over the range Osma, with 0. = 1, un lien li
expressed as
-
Jolay ) Jh10x)
xJ (ax) (ax)dx =
k )
(89)
1.16.
E.. !tlie. K'
Ole Chirr dolta and
o
l'or in = i,k.
The form of vie :0.12)ized Bessel riunction can then be written
u
(n)
12x1, (x/a)
(90)
Jila;)]
1:1107: (**'.
In particular, live
,1°) is witüen as
1.(0)
2x Jo, (maja)
(91)
[J, (aq)]
The nu:erical values for line normalized Bessel functions are to
b. found 1:1 'lablc XII :* Abilleix B, for the raiuc (XI.
An isometric Tranini udine nor!K:1:Iersel 'wictions of clo
similar ap!!:a.lS 11. 20:re: 3.

.
legali
-
:
:
1
...
-
-----
-
-
-
- .
.
.
.
.
:-
-
-
--
-
----
-
--
--
-
-
--
-
.-
imedim
--
-
-
--.-
-
-
.
-.
.-.-...
·
.
.--
:-
..--
.-
-
-
-
-
---...-
-.-
..
---
--- ---
::--
.
..-
:--
---
:
-
..
·
---
..
. .
.
.
.
"
-.
--
-
-
-
--
-
- -
--:-
Figure 3. The Normalized Bessel rinctions of Order Zero.
.
.
3
i
-
-
---
-
---
....
-
r---
-..
.-
.
.-
---------.
..-
.
---
--
-
---
-
-
--
-
-
---
-
-
-
diadal
----------
.
-
-.
-
-
....
:
;
-o..
.:-
1..
..
--.
----
.
.
.
.
.
.
.
.
.
-
-
CHAPTER III
CONSTRUCTION OF DEMONSTRATION MODELS
The construction of a model to display the Basel functions as a
function of the order, n, and radial displacement, X, as an uid to the
visualization of the quasi-periodicity 18 onc of the prir poues of this
dissertation. To this end, several criteria were established.
Because of the quasi-periodicity of the Bessei function, many
authors present a set of superimposed graphical or line drawings of
several orders of the Bessel functions on the game set of rectungular
axes with the result that only the characteristics of recularity o.nd
envelopment are readily apparent. A better method is that of an 18ometric
drawing of several orders of Bessel functions, but this generally
entails a lack of detail as to the relative magnſtile as a function of
the radial displacement.
It was felt that a three-dimensional model would best show the
form of the Bessel functions because, depending upon one's point of
obuervation, the superposition of several orders or an Issumetric view
of several orders could be seen.
The physical size of such a display was chosen so that several
desirable requirements could be satisfied. The size of the malei, len
inches by twenty inches, is small enough that it is portable enough to
use in a classroon and yet large enough that an averuve-sized class can
easily discern the principal details.
To satisfy the problem of stability and some degree of ixrmanance,
the base was made of a plywood sjab about three-fourtis of an inch thick.
32
33
The bases were chosen on the basis of appearance and size.
The wood 18
free of knots and therefore less susceptible to cracking and peeling.
'The representation of the orders and radial displacements were
thus established with the choice of the base. The form of the represen-
tation of the functions had several possible solutions. Among those
--
m
m
14
considered were molded plaster, molded plastic, plastic-of-paris, and
wood putty. These were discarded because it was felt that the model would
then be too heavy and bulky, in addition to lacking the possibility of
remuving a section of the model for closer study. Difficulty in the
fabrication and the pousibility of damage to the model were the reasons
for rejection of the possibility of inserting inctal strips between sections
of a plaster-filled contour plateau.
With the thought of a solid model still in mind, some consideration
was given to laying a series of inch square wooden strips parallel and
adjacent to each other. These wooden strips would have had holes drilled
into them and into these hules sections of metal rods could be inserted
with the heights of the rode proportional to the numerical value of the
Bessel function at that position. This idea, though it possessed merit,
was rejected due to the difficulty in clearly marking the rods to indicate
absolutely the value of the function at that position.
At this point, the method actually used vas evolved. Series of
parallel cuts were made into the bases tu be wed. Into these cuts
individual sheets of plexiglass were inserted.
These inserts were each
a representative of one order of the Bossel functiuns with the function's
value plotted along the vertical axis and the raial diuplacement plotteu
along the horizontal axis. Onto each sheet of plexiglass a grid was
scored so that the value of the function at any point could easily be
determined. The base of the sheets were shimed to insure a snug fit
into the grooves cut into the plywood platforms.
This particular form of model construction was felt to possess
several distinct advantages:
1.
The model size 18 quite accurate with regard to the form of
the Bessel functions as a function of order and radial distance.
The model is portable and can easily be dismounted for
moving and storage.
Because of the form of the display of the various orders,
selected parts of the display can be shown by removing the
other orders.
In the event that one of the sheets is lost or destroyed, a
replacement can easily be constructed.
CHAPTER IV
CALCULATION OF TABLES
The tables of the Bessel functions of the first kind for several
different orders are to be found in many places. Among these are some
textbooks of mathematical physics, some of the textbooks of differential
equations, mathematical handbooks, the Jahnke-Emde-Losch tables (3), and
in treatises on the Bessel functions.
The reason for their computation and tabulation here is that inany
of these tables list only the values for the zeroth and first orders, or
are so comprehensive that the problem becomes one of having too much
information with the result that one quickly turns to a set of tables
which are less comprehensive but more accessible. Also, most tab.lcs do
not include the values of the negative half-integral order Bcosel
functions beyond the first few.
The difficulty of extracting information regarding the numerical
values of the Bessel functions is similar to the difficulty of extracting
information regarding the theoretical development of the functions.
That is, one either has a great abundance of material or a remarkable
paucity of it.
To remedy this situation, calculations were carried out to determine
the numerical values of the various orders for integral values of the
radial components.
of the tables computed for this paper, only Table I of Appendix B
appears in the literature with any large degree of completeness. It was
included for completeness of the tables and because the calculations
35
carried out were used as a means of checking the accuracy of the methods
employed in determining the values in the other tables.
The calculations were carried out manually on a MonroMatic 16-213
desk calculator which has a 20x10 keyboard. The results were checked
against the values listed in the Jahnke-Inde-Lösch tables (3) whenever
such a comparison could be made. The differences, where they did occur,
were noted and the calculation was repeated to check for any error in
the mathematics. In addition, a sample of one third of all the other
values was recalculated as a check against error. In almost all cases
where there was some difference, the difference could be ascribed to the
extent to which the decimal fraction was carried in the calculations.
In this tabulation, the decimals were carried to ten significant figures
and rounded off to nine significant figures before continuing to the next
step of the calculation process.
It 18 felt by this author that any further calculations of this
type should be programmed for an autonatic computor. A rough estimate
of the number of mathematical computations and the rechecking felt to be
necessary indicated that an average of 9,600 individual operations were
performed for the determination of each orier of each of the Bessel
functions.
CHAPTER V
SUMMARY
The Besscl functions, though they are of great importance because
of their cylindrical and spherical geometry dependence, are not widely
known or used. It would seem that part of the reason for the lack of
vider application is due to the lack of wider dissemination of information
concerning their range of usefulness.
It is hoped that this paper will tend to help such understanding.
It is really not possible to expound upon the Bessel functions and their
uses adequately without going to the extreme of writing a tome which is
too unwieldy, but this short paper does try to show that the major topics
of derivation, explanation, application, and some numerical calculation
can be accomplished to some depth without sacrificing either clarity or
brevity
The topics discussed in this paper by no means exhaust the field
of the Bessel functions, even outside the highly abstruse area of the
subsidiary theorems and derivations. Yet the material covered is more
than adequate to give the student a basic grasp of the topic necessary
for the wide application of the functions in problems of mathematical
physics
The model constructed to prevent a visual display of the function
was designed to show the regularity of the function's contour plot and to
Give the student an idea of some possible uses of the function.
The form of the contour plot to some degree resembles the isunatric
display of the neutron cross-sections when plotted as a function of
37
38
energy and mass number. Other sections resemble the form of the pulse
height display of photon interactions in solids such as the alkai-halides.
The uses of the Bessel functions and their normalized solutions
are almost without number. The basic geometrical dependence ussures one
of their application wherever the problem involves either cylindrical or
spherical symmetry.
Some specific uses of the Bessel functions include those of wires,
coaxial cables, electrostatically charged disks, rings and rods, potential
fields of linear and circular accelerators, radiation fields of shaped
radioisotopes, heat flow and temperature gradiant problema, flow and ebb
of tidal estuaries, motion of real pendula, problems of flow in pipes
and along curved surfaces, and problems of mixed thermal and radiation
effects in atom'.c reactors.
BIBLIOGRAPHY
Byerly, William L., An Elementary Treatise on Fourier's Series
and Spherical, Cylindrical, and Dllipsoidal Harmonics with
Application to Problems in lathematical Physics. Ginn & Co
Boston, 1893.
Gray, A., and MacRobert, T. M., A Treatise on Bessel Functions and
Their Applications to Physics. Second Hition. Macmillan
& Co. Ltd, London, 1922.
Jahnke - Emde-losch, Tables of ligher functions. Sixth dition,
Revised by F. Losch. McGraw-Hill Book Company, Inc. New
York Toronto London, 1960.
MacRobert, Thomas M., Spherical Harmonics, An Blementary Treatise
on Harmonic Functions with Applications. Muthuen & Co. Ltd.,
London, 1927.
Watson, G. N., A Teacise on the Theory of Bessel Functions.
Cambridge University Press & The llacmillan Co., New York,
Second Hition, 1918.
HO
APPENDICI
APPENDIX A
TABLE I
EXPANSION OF THE INTEGRAL ORDER BESSML FUNCTIONS
Jg(x) = (2/x)J; (x) - 5 (x)
J., (x) = (8/- )J(x) - (14/x)5(x)
34 (%) = (i£, '32 - 6/x)JZ(x) - (24/7)J (*)
J3(x) = (38••* - 56x + 1)5, (x) - (21:47 - 11/x)J5 (x)
Jg(x) = (3, 09:0;* - 608/x + 16/%)J, (x) - (21:0/** - 64148 ) J. (x)
J7(x) = (-6, 680/3F - 7,530/x* + 2418/x2 - 1)J, (x) - (2,880/x* - 768/** - 24/x2 + 4/)J. (x)
J, (:) = (54,5, i20/x? - 111,350/x + 4,030/22 - 30/x)J, (x)
- (-:C, 320/x - 10,752/x* - 576/x2 + 8/)J. (*)
(2) = (7.0, 320, 920/22 - 1,27, 840/36 + 72,950/** - 728/x2 + i)J, (x)
- (345,120/x - 172,032/x* - 12,096/x* + 896/x + 24/x2 - 4/x)J. (x)
- 639,672,2.5017? - 3,095, 376/3€ - 258,046,'*+ 26,880,** + 1,008/** - 80/x)J (x)
TABLE II
the
XPANSION OF THE NEGATIVE HALF-INTEXRAL ORDER BESSEL FUNCTIONS
Joy p(x) = /2/ sx [ + cos x ]
Josis(3) = 12 *x [ • sin x - (1/x) cos x ]
(x) -12aXfr - (1 - 3./4) cus x + (31x) sin x ]
J.,(:) Sa RX [ + (1 - 13,'Xh?) sin x + (431x - 15/*°) cos x ]
Ju .(x) =V2/IX [ + (1 - 15/4 + 105/0) cos x - (10/X - .:05/x) sin x)
J-11 g(x):.. ! - (1 - 105/22 + 19.5/**) sin x - (15:48 - :20;X + 945/) cos x ]
J (x) = ax [ - (1 - 210.4.+ 12589 - 10,39; ) cos x + (21/x - 1,200/x + 10,30; Xo) sinux?
Jox) ..Tur + (1 - 370/4 + 17,325,4x4 - 135, 1377) sin x
+ (18/x - 3,50/300 + 12.570.'*" - 135, 135/8”) cos x)
J() = AXC + (- 2012? + 51,975/74 - 09:15,67.45.3 + 2,037,025 x") cos x
- (30;'* - »,930/x + 2'70,270,20 - 2,02'7,025/x?) sin x ]
J (x) = /27tx [ - (1 - 290/x+ 139,135.'*" - 729,723/xf. + 34,459, 425/xº ) si.: *
- (i76ix - 13.060 / x2 +2:15,9:15 / X6 - 16,216,200/x? + 34,459, 1:25. '*°) cos x )
J.:;**) - (1 - ., ****/2*2 + 325,315,'** - 18,918. 900 /4 + 310, 131,825/** -515:1, 72,075,449 ) cos x |
+ (55;'* - 245, 74:0/x2 +2,832,435,-91 891, 800/x? +054, 729,075/xº) sin x)
TABLE III
EXPANSION OF THE POSITIVE HALF-INTEGRAL ORDON BESEL FUNCTIONS
Jo;2(3) = 12/5x [ * sin x ]
Jz/p(x) = 1279x [ - cos x + (3/x) sin x ]
Ju 12(x) = /2771x! -(- 3/x?) sin x -(3/x) cos x ]
J.: g(x) = 12, "ax! + - 15/XP) cos x -(6/x - 15/*) sinx ]
Ja :-(x) = 2753 [ +: --:5;** + 205/** sin x +(10/- 205/x®) cos x ]
Jy /n(x) = /275X [ -(1 - 105/x + 945/x*) cos x +(15/x - 420/x3 + 9:45/x) sin x ]
J.Z. (*) = /27r* [i - 210,42 + it, 725'** - 10,395/8€ ) sin = -(21/x - 1,260/x? + 10,395/x") cos x ]
J.:;2(::) = 275x [ +1 - 378/x+ + 17,325/** - 135,135/f) cos x
-(28/x - 3,250/23 + 62,370/25 - 135,135/x?) sin x ]
une(z) = -2; 1x! +(1 - 630/4? + 51,975/** - 9:5,945/2€ + 2,027,025 /m) sin x
+(36'x - 5,530,30 + 270, 270/x . 2,027, 025,1x?) cos x ]
13:(::} = 277: [ - (1 - 990,432 + 135, 135/** - 4,729, 725/x + 34, 459, 125/**) cos x
+(1:5;% - 13,060/32 +0445,75/- 46,216,200/x + 34, 59, 425/*°) sin x]
(3- FX [ - ::35 + 325,315,'*. - 15: 9-2: 900,'xt + 310,134, 825/*- 5510, 729, 075/x0) sin x
-(55/- 25: 740/x + 2,532, U35,'* - 91, 691,600;':? + 55:-, 729,075/xº) cos x?
OS X
TABLE IV
EXPANSION OF THE NEGATIVE INTEGRAL ORDER SPHERICAL BESSEL FUNCTIONS
1-2(x) = [ +(1/) cos x ]
(x) =[-(-;x) sir. * -(1/) cos x ]
(x) = [-1.1: - 3;'*) cos x + (31x*) sin x ]
< (x) = [ +=;:- :5/x") sin x +(6.*- 15/X®) cos x ]
if(x) = [ + 1x - 05/X2 + 105/*) cos x - (10,'** - 105/x) sin x ]
3.-(x) ='(x - 105/x2 + 9.45/) sin x - (15/x2 - 1:20/** + 245/") cos x ]
3.(x) = [ -(:.!8 - 220/22 + it, 725; XS - 10,395/x?) cos x +(21/X2 - 1,260/x* + 10,395/x) sinx!
() -+: 'x - 370/x + 17,325/35 - 135, 135/x?) sin x
+(28/x - 3,50/** + 02, 370/x8 - 135,135/3) cos x ]
, (x) -1 +13 - 30 + 5:275/2 - , 945/x+ 2,027,0251x") cos x
-(30/x2 - .750 + 270, 270/8" - 2,027,025/3) sin x)
j (:) - [-(1.2 - 90,22 +23:.133/x5 ,729, 125/x + 34, 159, 25/x) sin x
-(-5/3 - 13,530,'** + 9.:5,945/46 - 16, 216, 200/XC + 34, 459, 1:25/xº) cos x ]
(x) = [ - *- -.-25/+ 3-5,3-5,4 - 18,918,900 jx? + 3.20, 134, 825/** - 554,729,075,272) 2052
+(55/32 - 25,7-0,* + 2,532, 835/x® - 92,391, 800/XP + 054, 729, 075.40€) sina
TABLE V
EXPAIISION OF THE POSITIVE INTEGRAL ORDER SPHERICAL BESSEL FUNCTIONS
t.
30(x) = [ +(18x) sin x ]
3,(x) = [ -(1/x) cos x +(1/**) sin x ]
js(x) = [ -(2/x - 3/*°) sin x -(3/8") cos x )
jz(x) = [ +(3./x - 15/xº) cos x -16/X2 - 15/**) sin x ]
3: (=) = [+1/x - 15,/X2 + 105/*}sin x +(10/22 - 105/**) cos x ]
jg(x) = [ -(1/x - 105/28 + 945/x) cos x +(25/32 - 420/x* + 945ix ) sin x ]
3-(x) = [ -(1/X - 210/** + 11,725/** - 10,3956x?) sin x - (21/X2 - 1,260/x* + 10:395/XF) cos x?
j_(x) = [ +(1/x - 376x + 17,325/** - 135,235/x?) cos x
-(28/x2 - 3,150/7* + 62, 370/x - 135,-35/x* ) sin x ]
je(x) = [ +(1/x - 630/x+ 51,975/28 - 015,945/x? - 2,027,0<5/x** ) sin x
+(36/x? - 6,930,'** + 270, 270/8% - 2,027,025/22 ) cos x ]
jo(x) = [ -(1/x - 390/2* + 135,-35% - 4,729, 725/x? + 36,459, 42;/**) cos x
+(5/X - 33, 360./** + 91:5:945/x- 16,216, 200; x + 3 is, 155, 125 x0) cer ]
3. (*) = [ -(1/x - 1,-185/x2 + 325,325/** - 18,918,900/x + 310, 134, 825/X2 - 554, 727,075;x2) sin x
-(55/*- 25,740,** + 2,832, 335/xf - 22,891, 505/x + 55->";29:0°5:***) cos x]
APPENDIX B
r
.
-
i
-.
-..
--.
-.
.
-.
-
... ...
... ... --

UNCLASSIFIED
ORNL
pop
R
41.
P
.
310
20F2
.
TABLE VI
BESS L FUNCTIONS OF POSITIVE INTEGRAL ORDER
(x)مل
(xكيل
0.00000
1.00000
0.75519
2.22339
(x)
0.0000
0.01955
005امد .6
م (
0
.
1289
ا
د ) -
2005. ۔
- 0.3271 :
0.00000
0.11490
0.35283
0.18509
0.369l
0.0555
-0.2267
-0.30l:
0.57572
0.33905
.0.0560
-0.32:57
-0.255
-0.0.
0.00000
0.0002
0.00703
0.0430
0.0320
0.25lll
0.3 20
(x)
0.00000
0.00247
0.03399
0.13203
0.2012
0.39123
0.3576i
0.15779
-0.0535
-0.26517
-0.21960
-0.01503
0
.
75-
د
ره:0
.
1
11. )
-۲
-م م
0
.
25
من
5.3 307
0.7i5
-0.02033
-0.2533
-0.17119
0
.
37
ون
c
.
i 577ن
0
.
12-
و
0
.
1
امهاا
0.25463
0.1390
-0.0693
ا
765 ق.م
ا
0.30905
0.3107
0.36183
0.1476
.0.16755
-0.29113
-0.18093
0.05837
0.22034
0.19513
0.00331
-0.17680
-0.1940
-0.0381+
0.1393
0.1832
0.0728
-0.09590
0
.
182
وما
0
.
2177-
با
0.21927
ا
ا ت ا " :: :
0
.
0762
با
ا ا:!:))
، ز) (
0.
23.
0.043 7
-0.17572
-0.223 224
-0.0707
0.1333
0.21510
0.0303
-0.097
-0.1- ..::
-0.10570
0.05
.. ( ۱
0.20592
0.17107
...
-0...
-0.122
..::
2.
... 02
-0.05503
-0.2306
-0.2382
-0.0737
0.3151
C.22037
0.13045
-0.0577
-0.1870
-0.15537
0.00357
0.15116
-0.i197
-0.2026
-0.15201
0.05l7
0.i8i9
0.25335
-0.00753
-0.15775
(.
0
.
107-
با
. ۱۱) ()
:: ۱۰
۱ از ۱۱۲) (۱)
0.06963
0.1805
0.13057
()
O
0
.
1503-
بن
6h
TABLE VI (continued)
به
ت :
(x)
(xكول ..
(x)و
(x)ميل
O 4 ( «
0.00000
0.00000
0.000l7
0.00000
0.00000
0.00002
0
.
0025
ا
0
.
000
ومن
0.00000
0.00000
0.00000
0.00000
0.00019
0.001
0.00695
0.02353
0.06076
11:50 - ) ،
J(x)
0.00000
0.00002
0.00120
0.0ll39
0.0908
0.131.0,
0.25 33
0.33919
0.33757
0.2043
-0.025
-0.2015
-0.
2 72
-0.1303
0.06:16
0.2021
0.16672
0.00071
-0.5595
-0.17275
- 0.05505
0.0izi7
0.05337
0.12958
0.23338
.3055
0.327.
0.2171
0.6i537
-0.17025
-0.2057
-0.15020
0.00402
0.0184
0.05 53
0.12797
0.22345
0.30505
0.31785
0.2297
0.0509
0.00000
(.00000
0.00000
0.00008
0.00093
0.00552
0.02116
0.05 892
0.12532
0.21 285
0.29185
0.30865
0.23038
0.0697
0.1l30
-0.22004
-0.1953
-0. 3
0.12276
0.00000
0.0000
0.00000
0.00000
0.00003
0.00035
0.0020
0.00833
0.02559
0.06221
0.123ll
C.2010
O
2 } ، {
-
را)
0
.
125
و
0
.
207
قيا
0
.
280
جها
0
.
270
دا
N
سم 1،
.
.
0
.
1
:20ما
0
.
03
کا
0.29266
0.2357
0.0999
-0.06622
م
ir16
r
ح (
،
0.3007
0.23378
0.08500
-0.09007
-0.20620
-0.199il
-0.07316
0.09155
0.186
-0.23197
-0.17398
-0.00702
0.15372
0.19593
0.0929
-0.07355
0.
15.
0.1675:-
0.05133
-0.2167
-0.3
0
.
1913-
و
2005. ۔
م
()
0
.
197
ما
0.12512
-0.09837
2.05235
50
.
TABLE VII
BESSEL FUNCTIONS OF NEGATIVE INTEGRAL ORDER
م
و
ح
)
و 24
لد
(x)
(xكيل
(x) ل
(x) ل
(ع)
لها
0.00000
0
.
0002-
ما
0.00000
-0.1: 005
-0.577
-0.33905
0.06044
0.32757
6.27556
0.00oo0
0.2:20
C.35283
0.303
0.00000
-0.01956
-0.1289
-0.30906
-0.3107
-0.3 83
-0.
17
.. 55
c.2012
0.18093
-0.05837
0.00000
0.00002
0.000.20
0.013
0.0 ::05
0. 0.
0
.
3
غدود
رد:0
.
6
0.00000
0.0027
0.03399
0.3203
0.28112
0.39123
.375
0.179
0.10535
-0.2547
-0.21050
-0.02503
0
.
25
خ
0
.
00 37:با
-0.2352
-0.2531
0.3319
0.33757
0.2013
0
.
12
ماما+
-0.00703
-0.0302
-0.13203
-0.26lil
-0.36208
-0.34789
-0.18577
0.05503
0.23 106
0.23828
0.075447
-0.13164
-0.22037
-0.13015
0.05747
.
0
.
0
"ات
.
0
.
0i
وان
-0.21+227
-0.36
-0.1i99
0.25463
0.1390
-0.0893
-0.21775.
-0.1520i
0.05l7
0.1839
0.17578
به 24
با2273. 6.
0
.
223
ماما
و i82.0
-0.19513
-0.0033
0.17680
0.2: :27
-0.20:52
-0.21372
-0.1i803
0.0611
0
.
072
نا
م اما لن
0.07031
-0.13337
-0.210
-0.05039
0.057:56
0
.
1
ومن
.i197
مت0
.
20
با2025. -
0
.
15
دوت
* C110.
0.038
-0.133
-0.1332
با1870.)
از) لا
6
.
187
ود
-0.00757
-0.1577;
0.10570
0.16672
0.00071
-0.15595
-0.1767
-0.05505
...071
0.0563
c.io
C.3057
0
.
072-
عن
0.15537
-0.00357
-0.15115
ژنره..
0
.
1603
من
0.05 90
5
TABLE VII (continued)
x
J_(k)
ܐܬ݁ܨ.
ܐ
12
ܐܐ
-9
6.ooo
6.5OK
6.006
o.com
6.00
ooooo. ܘ
.ܘ
-0.0003.?
56ܗܣ.ܘ
0pos.ܘ
Sܪܐܘܘ
ܡܣܘ.O
com.ܘ
ooo8. ܘ-
oooo.ܘ
comܘ
o.mo
o
.
co25-
ܛ:
o .opic
-o.o1517
-o.05337
-c.2956
O.OOOOO
-o.ooop?
-6.0035
o.ooo93
-o.co552
ܐܬo18.ܘ
05553. ܘ
16ܐܣ.
o
.
o020-
ܪܐ
23358. ܘ-
o.12797
c-223;5
0:3050-
-..05892
-6.12632
-0:32058
-0:32745
-6-21673
-o.;3817
0
:
21 88ܪܐ
.ܘ
pl
Opis.ܘ
6
.
6 6ܬܐܐ
695.ܘ
o2323.ܘ
5O72.ܘ
0
.
120 69ܙ
2078:ܘ
ܦܪܐ28o.ܘ
7ܪܐ30:ܘ
23378.ܘ
0850.ܘ
-o.co833
-o.2559
-o.06221
co،12311
woo2010
85;31:ܘ
4
o:221:97
;.oi,509
-o.29185
-0:30385
-0-23038
-o.96597
ܐܕܐ276-0-
ܪܐܘܙܐܐ،O-
7025ܐ.ܘ
، ܘ
23
:
057
Sodo܂.ܘ
6+ܐܪܐ03..-
.25ܝܵܐ.o-
-o2258
-0:2357+
4 4
u
r
{
{
rH rܓ݁ܙ 4
-
ivo fuoܘ 4
;
ܘ3ܐܪܐܘ
-023197
-0:37398
-..0702
O:22ool}
-o:-8754
15378.ܘ
953ܐ. ܘ
285ܪܐo.ܘ
-o.907
-0:20626
-.1911
-..0315
05139. ܘ-
|ܪܐ6ܐܐܘ
22-&، ܘ
o:19593
-..0738
ܗ. ܘ-
95
06822. ܘ
19139، ܘ
06ܝܐ20،ܘ
9837.ܘ
06135. ܘ-
_
6
.
0929
.ܐ
0
:
39-
ܐ7ܬܐ
9155.ܘ
-o.12512
TABLE VIIL
BESSEL FUNCTIONS OF NEGATIVE HALP.. INTEGRAL ORDER
و
و
کچ
کو
ل
د
||
0.3110
کا075، 90
عرار، ,ا۔
0
.
23-
اما
2.87635
0.55231
0.05591
0.16196
5 ،030
3-1 )
.13 .2796
1 .6795
.0.7c75
-0.379
.
1 ،102t7
-0.39577
0.0871
6.3670
0.3219;
1
.
26 13و
0
.
62
ها
. 6
3 ،1032
.5 .5605
-0.25997
c.10122
C 3:27
( .22735
580ما2. 6.
ما
07، 1.
TiTژ، .
036، 0۔
0.3396
ح ا
0
.
171
من
0.23795
6
.
0
م16
6
.
322
ها
04105. ر)۔
-0.23061
-0.27397
-0.0263
-0.2013
-0.0500
0.13031
0.1989
0.10701
-0.0639
0110
0.06725
-5 .4053
-0.2101c
.6 ،319
.6 ،
012
6.
18
6 227
6.22575
.6 .1938
.0.3687
-0.002
0.
00
(2006
0.230
5
.
158
اما
7اما2، 6
و به تو
عمان.6
0.108
0
.
17-
های
02523، 6۔
0
.
16
ام
0
.
10.
ماهان
. .21232
-0.2li70
0.01o7
C .19:33
0.20091
0.0859
-0.1517
6 .1916
.0 .05324
0.12119
وق.)
-0.13301
0.2202
0.7 1926
0.2021
-0.077003
.0 ،12177
0.21027
-0.12332
6.1756
0.2212
0.12
0.0613
0
.
1
ملالها
6
،
19
ارنا
وهم ها
-0.21336
-0.12922
0938
6.18918
0.1333
.6.03697
-0.16657
1250i، 6۔
1950، 6۔
0.01i01
.و
دمام
6.13
0.
01
0
.
1i13-
ما
5 ،10298
.6 ،09361
0.
00
قوم1.)
.0.
107
0
.
15
آنها
3.07200
بابا037. ر)۔
0
.
173
عبا
6
.
14
بادبا
مها 01. 6.
.6.1716
TABLE VIII (continued.)
ل
ع
(
(علي
علي
(x)
.192
۔
21
/
2
.1372
8,665.616
,ا۔
20
.
356
قيا
9
,
372
.
60
28, 316, 562.2i
.696.2731
) ,2ا۔
53
.
1991
وا. 051 ,339 ,535
3
.
7i197وا ,ی
T552.مناه
من 3635..
5,151.700
193.55935
2.57990
.5725
1.5910
-768.79121
.85.35104
.13.7939
38 .31507
10.1170
3.79639
6.929
0.5329
0.306fo
.3.0
.0.13055
دورio.گا
0
.
763
مبا
3
.
55655۔
0
.
795-
و
..26085
0.7kT
0.29093
9.77282
2.5175
1.16793
0.6563
...6330
.0.33232
.0 ،50036
-0.29930
-0.0926
0.10725
0.286
0.25032
0.1670
.0775
-0.
265
0
.
2-
نن
صباح...
0
.
05-
وفي
0
.
12
موبا
-0.45337
.0.28360
-0.10955
وهاها.0
0
.
28
واها
.0.08780
ه
0
.
27
0
.
073
ا
.0.25013
-0.1001
0.03358
0.235
0.2479
0.11659
.0.058s
00:22
0
.
16 63با
0
.
228.
م
20:15
0.22059
c.15910
0.01o30
-0.152:
..1&2
..0553
01:9
مع1. .
0
.
036-
ت:
0
.
28
ويها
0.3097
2.9olio
0.105
0.1979
0.0751
-0.0350
.6.1688
-0.003
-0.1666
-0.20193
-0.10676
0.05623
.0.12265
.0.0.5
0.iST:
2.20020
0.3120
ا5
مم
TABLE IX
BESSEL FUNCTIONS OF POSITIVE HALF-INTEGRAL ORDER
J
(x
(x)
(x)
1/2
372
0.00000
0.671110
0.51301
0.06502
-0.30192
-0.34217
-0.09101
0.19812
0.27906
0.10961
-0.13727
-0.24055
-0.12358
0.09297
0.21124
0.13397
-0.05741
-0.18305
-0.14123
0.02743
0.16286
0.00000
0.24031
0.49129
0.47772
0.18529
-0.16965
-0.32793
-0.19905
0.07593
0.25450
C.19798
-0.02293
-0.20466
-0.19366
-0.011:07
0.16543
0.1874
0.04230
-0.13202
-0.17953
-0.05767
512
0.00000
0.04950
0.22393
0.41271
0.44039
0.24038
-0.07292
-0.28343
-0.25062
-0.02477
0.19666
0.23131
0.072442
-0.13767
-0.21423
-0.10088
0.09257
0.19351
0.11923
-0.05578
-0.17258
0.00000
0.00719
0.06851
0.21013
0.36582
0.1002
0.26712
-0.00340
-0.23256
-0.26830
-0.00966
0.1294)
0.23484
0.14071
-0.062:44
-0.19909
-0.15850
0.01561
0.16513
0.16485
0.02150
0.00000
0.00081
0.01589
0.07761
C.19930
0.33366
0.38461
0.28002
0.04712
-0.18388
-0.266440
-0.1519)
0.06457
0.21344
0.18300
0.00798
-0.16192
-0.18749
-0.05500
0.11652
0.18011
J (x)
11/2
0.00000
0.00007
0.00295
0.02265
0.09262
0.1905€
0.30980
0.36345
0.28557
0.08436
-0.14013
-0.25374
-0.186l: I
0.00705
0.18010
0.20386
0.05741
-0.11386
-0.19255
-0.10966
0.05953
14
1
.7
.
hu
h
"
21
TABLE DX
(continued)
.
1572
192 .
8
13/
0.00000
0.00003
0.00047
0.0052
0.02787
0.08556
0:18532
0:2931
0.34555
0.28739
0.11228
-0.20180
-0.23543
-020747
-0.04252
0.74251
0.2085
012362
•0.0673
-0.38000
-0.373
0.00000
0.00000
0.00005
0.00112
000792
0.03210
0.08743
0.27733
0.27359
0.33019
0.2809
0.23345
0.0505!
9.1452
-0-2336
-0.08221
0.10180
0.20092
3•14735
•0.0338
-0-25531
J (x)
17/
0.00000
0.00000
0.00001
0.00022
0.00296
9 •01022
0.03520
0.08852
0.17385
0.26311
0.51685
0.28377
9.14962
-0.0004
•0.1277
-9.22271
-0.1128
0.06345
0.28553
9.16936
0.03089
0.00000
0.00000
0.00000
0.00003
(0.00044
0.00288
0.0233
0.03785
0.0891
016727
0:25:56
9.30530
0.28054
0.16212 .
-0.015
•0.1721
•0.228
•0.1375
0.02737
01650,
018154
0.00000
0.00000
0.00000
0.00001
0.00007
0.00071
0.00383
0.03422
0.0003
0.08957
0.16301
0.24305
0:29466
027701
0.37183
0.00567
-0.15042
••2170
0.15612
•0.00434
0.14163
12.
0 ,Ur
0 0
20
9g
*
.
""
"
"""
WI
TABLE X
SPHERICAL BESSEL FUNCTIONS OF POSITIVE INTEGRAL ORDER
(x)
3 (x)
ano.vifunt
0.30118
0.13539
0.34562
0.11611
-0.09503
0.84148
0.1:5464
0.0=705
-0.19220
-0.19179
-0.04657
0.09355
0.123:56
0.0.1579
-0.05440
-0.09090
-0.014:???
0.03232
0.07076
0.01:335
-0.02739
-0.05655
-0.04172
0.00789
0.01565
0.06204
0.19845
0.29864
0.27629
0.13473
-0.03731
-0.13426
-0.lllc5
-0.01035
0.07794
0.08854
0.02620
-0.04786
-0.07176
-0.03265
0.02900
0.05882
0.03522
-0.01604
-0.04831
-13.09429
0.03365
0.10632
0.07847
-0.00866
-0.07405
-0.06732
-0.00472
0.05353
0.05073
0.01285
-0.03900
-0.05162
-0.01312
0.00901
0.06072
0.15205
0.22924
0.22982
0.13668
-0.00261
-0.10305
-0.11209
-0.00383
0.04891
0.08497
0.04891
-0.02092
-0.06443
-0.04966
0.00444
0.04878
0.04740
0.00.03
0.00102
0.01408
0.05616
0.12489
0.18702
0.19679
0.13265
0.02088
-0.07682
-0.10558
- 1.05742
0.02336
0.07419
0.06130
0.00258
-0.05073
-0.05699
-0.01625
0.03350
0.05048
0.00007
0.00261
0.01640
0.05177
0.10681
0.15851
0.17217
0.12654
0.03525
-0.05551
-0.09562
-0.05905
0.00245
0.06033
0.06597
0.02112
-0.03461
-0.05691
-0.03153
0.01668
11
rii
L-Mount
1
A
.
.
TABLE X
(continued)
1 x
x
0.00000
0.00000
0.00002
0.00028
0.00161
0.00631
0.01793
0.03953
0.06984
0.00001
0.00042
0.00397
0.01746
0.04796
0.09320
0.13790
0.15312
0.11998
0.04450
-0.03847
-0.07458
-0.07212
-0.01390
0.04579
0.05520
0.03454
-0.01853
-0.05176
-0.04129
0.00000
0.00005
0.00081
0.00496
0.01799
0.04472
0.08391
0.12123
0.13794
0.11339
0.05042
-0.02175
-0.071157
-0.07324
-0.02628
0.03190
0.06107
0.04353
-0.00338
-0.04353
0.00000
0.00001
0.00016
0.00123
0.00573
0.01801
0.04193
0.07614
0.11000
0.12558
0.10723
0.04740
-0.01392
-0.06457
-0.07207
-0.03536
0.01929
0.05481
0.01:870
0.00865
0.10010
0.11529
0.08891
0.05635
-0.00517
-0.05540
-0.06946
-0.04178
0.00823
0.04745
0.05088
0.00000
0.00000
0.00001
0.00004
0.00040
0.00196
0.00674
9.01774
0.03742
0.06461
0.09192
0.10661
0.09629
0.05756
0.00190
-0.04713
-0.06599
-0.04612
-0.00125
0.03969
.
TABLE XI
SPHERICAL BESSEL FUNCTIONS OF INEGATIVE INTEGRAL ORDER
()
-1
0.54030
-0.20794
-0.33000
-0.16292
0.05673
Orl6003
C..0770
-0.01019
-0.10124
-0.00390
0.00040
0.07031
0.05920
0.00958
-0.04210
-0.05985
-0.01613
C.03659
0.05204
C.02040
iz) IDO
-1.39177
-0.35074
0.052'36
0.23036
0.13045
0.07321;
-0.1092
-0.1211:0
-0.03454
0.06280
0.09087
0.03385
-0.03769
-0.07147
-0.01:162
0.02174
0.05751
0.03952
-0.0106)
-0.06667
1 (x)
-5
112.89559
11.446231
0.91835
0.39142
0.1E352
0.02794
-0.0516:.
-0.13509
-0.02333
0.06776
0.07573
0.09324
0.02815
-0.040945
-0.06304
-0.03864
0.01037
0.05405
0.04156
-0.00421
3.50197
0.57309
0.0..701
-0.2012
-0.:37??
-0.10330
-0.02132
0.05774
0.03275
0.04241
-0.02509
-0.05392
-0.04623
0.00764
0.0:4630
0.0.152
0.00335
-0.03282
-0.01:257
-0.01049
Intrigrirt retri
-IĆ.04308
-1.3439
-0.50302
-0.21804
-0.02545
c.12175
0.15273
0.0815€
-0.02810
-0.09533
-0.07942
-0.00551
c.oblig
0.06942
0.02266
-0.03917
-0.05928
-0.02765
0.0238€
0.05002
-532.30412
-18.54725
-2.2470.2
-0.65266
-0.320:16
-03636ć
-0.03.-5
0.070,0
0.11899
0.09333
0.01746
-0.0531:0
-0.03068
-0.04313
0.0181é
Dnia Gin F
0.00030
0.04955
0.00003
-0.04436
-0.04817
19
20
.
TABLE XI (continued)
(عل
_(ع)
(x)
-lo
.
-7
:0, 29.79
.
-14, 05.12
.5l7.055T
65, 339.978
,35.
489
,
685
بار.
,203 ,672
و923
.
6
97
.
79i8
و
( اا::
را
530
.
126
مبل
باi 0
.
05975
29
.
750
و
.37, 802.3
.556.29&ei
365, 361 .7
4,698.8796
3.96776
s85قها. 3.
مند. ا2
7.32072
2.3790i
6.5163
2.27210
c. 28
0.03329
1
.
23۔
:: :
13.52320
2.56386
0.7962
0.36463
0.21038
0.12156
0.0950
.0.2590
-0.3703
-0.03262
.6.0zli
.7.68970
.1.81976
-0.59727
26.96380
5.00039
.35090
0.5-752
.:C
0
.
31.
ژلها
0
.
05.
ا
.6.189
0
.
27
جانيا
0
.
0 251یا
.0.1086
.0.9319
-0.03106
03318.ن
مايد.0-
0
.
01-
ما
0.17514
c.10750
م
18عام.)
س
خدما0
.
0
م
0.07ks6
0.05472
-0.00323
م
6.1077::
0.09057
0.0057
-0.0253
-0.05559
-0.05828
.0.05092
6.03559
0.0575
0.058:
0.07381
0.082T
0.05327
-0.00320
-0.0615
-0.08695
-0.04690
2.olo87
0.05787
(.06013
0. 220
-0.02586
-0.05181
-0.02035
-0.0660
-0.07857
-0.05398
-0.00500
م
م
0
.
0 915با
م
0
.
00-
ماها
0
.
05-
اماما
0
.
063-
با
0
.
055
وما
م
.0.0965
-0.03070
0.0i576
ة
c.03050
0.05755
0.03577
تحت 0.c
200
Whe
S.
N.
1
TABLE XII
NORMALIZA BESSEL FUNCTIONS OF ORDER ZERO
00)
(0)
(0)
(0)
x/a
0.00
0.10
0.20
0.50
C.!,0
0.50
0.60
0.
oo
1. 6:16
2.22.363
2.51337
2.5971
2.42604
2.20901;
1.73903
1.25933
0.14463
0.00000
0.00000
3.57513
3.92995
2.83178
0.81795
-1.45405
-3.28109
-4.11313
-3.70242
-2.17607
0.00000
0.00000
4.98331
3.262!1
-0.99796
-4.514:57
-1.83354
-1.678714
2.66843
5.12793
3.96337
0.00000
0.00000
5.53744
0.28521
-5.51609
-4.37€84
2.23400
6.07043
2.36660
-4.27470
-5.61802
0.00000
0.00000
5.40901
-3.62442
-5.90877
2.99766
6.35178
-2.05652
-6.66510
1.04667
6.82552
0.00000
0.70
0.10
0.96
1.00
.
.
T
IL
TABLE XII (continued)
(0)
(0)
x/a
0.00
0.20
0.20
0.30
0.40
0.50
0.00000
4.26341
-7.05677
-0.71397
7.11bở77
-2.80397
-6.121:33
5.70253
3.143997
-7.58870
0.00000
0.00000
2.30236
-7.80053
6.09990
1.37529
-7.56244
6.56743
0.67566
-7.27505
6.94938
0.00000
0.00000
-0.26083
-5.291440
8.52275
-7.74493
3.28719
2.711226
-7.46730
8.63059
-5.67610
0.00000
0.00000
-3.163646
-0.20073
3.67055
-6.63083
6.60622
-9.28972
8.56952
-6.55138
3.54016
0.00000
0.60
0.70
0.00
0.90
1.00
-
DATE FILMED
10/31 / 64






$
4
*
1
.
.
*
...
- LEGAL NOTICE
This report was preparod as an account of Government sponsorod work. Neithor the United
Suatos, nor the Commission, nor any porson acting on baball of the Commission:
A. Makos any warranty or reprosontation, exprossod or implied, with rospoct to the accu-
racy, complotonoss, or usofulness of the information contained in this roport, or that the wo
of any information, apparatus, mothod, or procos, disclosed in this roport may not infringo
privately ownod rights; or
B. Asrumos any Ilabilities with rospoct to the use of, or lor damagoi rosulting from tho
use of any information, apparatus, method, or procons dioclorod in this report.
As usod in the abovo, "person soting on bohall of the Commission" includos any om-
ploy.. or contractor of the Commission, or employee of such contractor, to the oxtont that
such employee or contractor of the Commission, or employee of such contractor proparos,
disseminatos, or provides accous to, any Information pursuant to his omployment or contract
with the Commission, or his omploymont with such contractor.

7
E
nd
END
I.
1
.
X2
.
.
L