Table Of ContentISSUED BY THE COMPUTER CENTRE OF THE ACADEMY OF SCIENCES, U.S.S.R.
TABLES
OF
WEBER FUNCTIONS
VOLUME I
by
I. YE. KIREYEVA and K. A. KARPOV
translated by
PRASENJIT BASU
PERGAMON PRESS
NEW YORK • OXFORD • LONDON • PARIS
1961
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The translation has been made from the book by I. Ye. Kireyeva and
K. A. Karpov entitled Tablitsy funktsii Vebera* Volume I, Moscow, 1959
Library of Congress Card Number 61-12444
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PREFACE
THESE tables are among the first to be published, which are devoted to Weber functions or functions of a
parabolic cylinder. The calculations for the tables were carried out with a high-speed "Strela" electronic
computer. The calculations were checked and the manuscript prepared for publication under the super
vision of L. S. Bark and R. D. Vedeshkina, our colleagues at the Computing Centre of the Academy of
Sciences, U.S.S.R. The space diagrams were drawn by T. V. Firsova, senior technician and laboratory
assistant. The authors express their gratitude to these colleagues for assistance rendered in the preparation
of the tables.
VIÎ
INTRODUCTION
THE main difficulty in solving the wave equation
a2o d2a> d2o> .
f A
;+ r+ r+ fcO = 0
dxi dx dx 2
2 3
in parabolic coordinates by the method of separation of variables consists in finding solutions of ordinary
differential equations of the type
££ W (, >_^ .o (i)
dze2 + Vx+ ' 2 D4 f(x) k
d2W (z) / ^ 1
p L
d.» + ('+-J + 7) "VW " "• ,2)
Equations (1) and (2) are called Weber's equations, and their solutions Weber functions or functions of a
parabolic cylinder [1] - [4].
The wave equation, expressed in parabolic coordinates, occurs in quantum mechanics, radio physics,
aerodynamics, hydrodynamics and other fields [5] - [7]. The differential equations (1) and (2) can be used as
standard equations for deriving asymptotic forms of solutions of an equation of the type
y" + \X2 x2 r (x) + q (x)] y = 0
2
for large X and x, which varies in the finite interval (0, /) [r (x) is finite and q (x) is a finite and differentiate
2
function]. They can also be used for studying the asymptotic laws of distribution of the eigenvalues of this
equation [3], [8].
Equation (1), as well as equation (2), is considered in detail in the work [3], and the solutions of equation
d2 W (a x) (x2 \
(2) written in the form T2—1 + ( « )W(a, x) = 0 are tabulated for real x and integral values of a.
dx2 \4 )
Since equation (2) is reduced to equation (1) on replacing p + {- by i(p + i) and z by ze~l(7r/4), and there
exist the relations
W (z) = D . ^ ^ C z e ^ ), (3)
p
D(z) = lf_ , (ze-i(n (4)
p i + p+i)
between the solutions of these equations we shall confine ourselves below to a consideration of equation (1)
only. For this equation the functions D (z), D ( — z), D-p-^iz) and D^ _ ( — i z) for p^O, 1, 2, are
p p p 1
linearly independent solutions in pairs.
In the present work the solution of D (z) is tabulated for z = x(l + 0 and real p is satisfying the initial
p
conditions
pW r[(i- )/2]' pW ix-p/2)'
P
These conditions are selected by proceeding from a representation of the functions D (z) by means of a
p
degenerate hypergeometric function (see formula 15). Since x takes both positive and negative values, two
linearly independent solutions D [x(l + 0] and D [ —x(l + i)] are tabulated. By using the principle of
p p
ix
x Tables of Weber Functions
symmetry (p. xi) and the recurrent correlations (p. xxi), it is possible to calculate another pair of linearly
independent solutions of equation (1).
The functions D (z) are single-valued, analytical over the entire z-plane; all their zeros lie on the real
p
axis [4].
Weber's function D (z) can be expanded in series of the type
p
-( /2)-i - »/4) « (_)™r^( _)]
2 P (2 1 r /w p
for p / 0, 1, 2 . and
£> (z) = 2-<"/2>e-(-'8/4' Y J^Ü-Pl-(z /2)''-2(- (6)
v
P k = ok\(p-2k)\ N
forp = 0, 1, 2
For the second case it can easily be established from (6) that
D(z) = (-l)'D (-z).
p p
The asymptotic series given below describe the behaviour of Weber functions for large values of the
modulus z [4]
D(z) ~ e-<'-'4v{l - P(P~ 1) + P(P-0(P-2)(Pz3) |
p 2
3rt ^ 3TT ,„.
< arg z < —, (/)
4 4
D (z\ ~ e-(^4) z" Il - p(p-1) + P(P-l)(P-2)(p-3) ) _
pK) X 2-z2 2-4-z4 J
_ V(27t) ,. _ _! L (p+l)(p + 2) (p+l)(p + 2)(p + 3)(p + 4)
el feUV4) z p
T(-p) \ 2-z2 2-4-z4
7T 571 ,„.
- < arg z < —, (8)
4 4
D (z) ~ e-(--s/4»z"Il - p(p-1) + P(P-J)(P-2)(P-3) I _
p() I 2-z2 + 2-4-z4 j
V(27t) _ „. „_ _ f (p+i)( + 2) (p+l)(p + 2)(p + 3)(p + 4) _
c p c(zV4) p 1 t p
r(-p) j 2-z2 2-4 z4
5JT 7i
< arg z < .
4 4
There are various integral representations for Weber functions. These can be found in [2], [3] and [9].
The following recurrent correlations connecting the Weber functions with their derivatives may be found
useful when the tables are used :
D (z) - zD (z) + pD .,(z) = 0, (10)
p+l p p
-- D (z) + izD (z) - pD ^{z) = 0, (11)
p p p
dz
f Z>„(z) - \zD {z) + D (z) = 0. (12)
p p+l
dz
Tables of Weber Functions xi
The functions D {z) of a parabolic cylinder are connected with a number of special functions. We give a
p
few formulae :
DJz) = tl2l£ z-iM(p/2)+i..x[-) + l 2) z--M(p/2)+i.x [-], argz <(3TI/4). (13)
p r[(i-p)/2] (p/} *• 4V2/ r(-p/2) (p/2)+-4^ y ' '
2
D (z) = 2^) + -z-^ ,_ (z2/2). (14)
p (p/2)+; i
Here M (z) and W (z) are solutions of Whittaker's equation [4].
fc>wl km
D,(z) = 2""» W V* / £. 1. z!\__£^) (tlP. 3. ?!\l
e F f (15)
Irca-p)^]1 v 2'2' 2; n-p/2) V 2 ' 2' 2/j
where F (a, 7, z) is a degenerate hypergeometric function.
i 1
When/? = n is a positive integer, the Weber functions can be expressed by means of Hermite polynomials
D„(z) = 2-<»/2>e-<-^>//„(-^\ H„(Z) = (-1)"e<2 ^(e"'2), (16)
and when/? = — « is a negative integer, by the probability integral
O(z) = — e *~dx.
oo
As mentioned above, the function ^(tf, x) is considered in the work [3]. Using relation (3) and bearing in
mind that a= —p — \, we find
D. . (xe'<*'4)) = W(a x). (18)
i+Ifl 9
For information on tables of functions by means of which D (z) can be expressed, see [10]. Detailed data
p
on the properties of functions of a parabolic cylinder are given in the works [l]-[4].
CHARACTERISTICS OF THE TABLES AND METHOD OF CALCULATION
The tables given contain values of the real and imaginary parts of the functions D (z) = u (x) + iv (x)
p p p
in Cartesian coordinates for z = x (1 + 0 and real/?.
The variable x varies from —10 to +10 in steps of 0.01 ; the index p takes values from 0 to 2 in steps
of 0.1 for | x |^5.00, and in steps of 0.05 for 5.00<| x |<10. The error in the tabular values of the functions
does not exceed 0.6 of a unit of the fifth decimal figure for all the remaining values of x and /?.
Because of the principle of symmetry [11], there exists for the functions D (z) the relation D (z) = D (z),
p p p
which enables the given tables to be used for finding the values of D \_x(l — /)].
p
The diagrams 1-10 give a general idea of the nature of the curves w (x ), v (x ), up (x), vp (x) and the
p 0 p 0 0 0
surfaces u (x) and v (x).
p p
The steps of x and p in the tables are so chosen that, depending on the region, three to six tables of
values have to be used for interpolation with respect to x and six to eight for interpolation with respect to /?.
9
The tables were calculated from asymptotic series by the method of piecewise approximation of the
solution of differential equation (1) by polynomials in z. For the functions u {x) and v (x) we get from the
p p
asymptotic series (7) and (9) :
«,(*) = (*V2)' U cos {^ -7f] + ^2 ™ fê - nf)\
v (x) = (*V2)P U cos f |. - ^j - A, sin f| - ^
p 2
Xll Tables of Weber Functions
Up(x0)
tooj-
50
100
Fig 1. Fig 2.
vp(K0)
.,
!6
12
X„=-4. /"" '
\
b
X y --? \
4 . \
^ >" x,.-0 \
^->^ ^ .,.."' \ ,, , . x A . ,
- -=ü^^rp_?__ _____ !-0 1-5 N\2-0
-4 ^^^-^ "" v
^ ^ - - ^^ xo=2
-8 V
v« -- *-* -50L
Fig3. Fig 4.
Tables of Weber Functions xi H
Tables of Weber Functions
XIV
¥m 7.
Fig 8.
Tables of Weber Functions xv
Fig 9.
