Tables of Lommel's Functions of Two Pure Imaginary Variables

286 Pages·1965·22.545 MB·English

by  L. S. Bark and P. I. Kuznetsov (Auth.)

TABLES OF
LOMMEL'S FUNCTIONS OF
TWO PURE IMAGINARY VARIABLES
by
L. S. BARK and P. I. KUZNETSOV
Translated by
D. E. BROWN
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS • FRANKFURT
1965
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Copyright © 1965
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Library of Congress Catalog Card Number 63-18940
This translation has been made from a book by L. S. Bark
and P. I. Kuznetsov entitled TaÔAUi{bi i{UAUHdpunecKux
4>yHKu,uü or deyx MHUMUX nepeMennbix (Tablitsy tsilindri-
cheskikh funktsii ot dvukh mnimykh peremennykh), published in
1962 by the Computing Centre of the Academy of Sciences
of the U.S.S.R., Moscow
Printed in Poland
FOREWORD
THE PRESENT tables are among the first to be published on cylinder functions
of two pure imaginary variables. They were computed on the "Strela" electronic
computer. The computations were checked and prepared for the press in the Ana
lytic Machine Department under the direction of R. D. Vedeshkin. The figures
were drawn by chief Laboratory Technician T. V. Firsov.
The authors express their gratitude to scientific associates N. M. Burunova,
K. A. Karpov, and A. I. Sragovich for their assistance during work on the tables.
The Authors
vii
INTRODUCTION
LOMMEL'S functions of two pure imaginary arguments are defined by the
Neumann series
(i)
(2)
where I (x) are Bessel functions of an imaginary variable and integral index
n
(modified Bessel functions). The functions Y„ and 6 were introduced in 1946 by
n
P. I. Kuznetsov [1] in connection with an investigation of the transients in a semi-
infinite homogeneous electric transmission line (Lecher system) when constant
unit voltage is switched in with zero initial conditions.
Numerous problems of mathematical physics reduce to a solution of the telegra
phist's equation with various initial and boundary conditions [2, 3]. Examples
include: (1) investigation of transients in electrical circuits [4-7]; (2) analysis of
non-stationary processes in the theory of heat conduction [8] ; (3) the theory of the
laminar motion of a viscous fluid [9], etc. The solutions of these problems are usually
expressed in terms of indefinite integrals of the form
(3)
(4)
(5)
with the aid of the Poincaré-Picard method [2, 3]. It was shown by Kuznetsov [1]
that the indefinite integrals (3)-(5) are expressible in terms of the functions T (y x)
n 9
and 0 (y, x).
n
ix
x Tables of LommeVs Functions
The integral
(6)
or varieties of it are encountered when solving the following problems: (1) inte
gration of the two-dimensional normal probability density over the area of a circle
whose centre is displaced relative to the mean of the distribution [10]; (2) distri
bution of the envelope of a sinusoidal signal in the presence of Gaussian random
noise [11]; (3) investigation of certain limiting distributions for the number of times
one of the states of a two-state homogeneous Markov chain is hit [12]; (4) inves
tigation of temperature fields with instantaneous heat sources [8]; (5) solutions
of problems arising in studies of heat and mass exchange phenomena in the presence
of phase and chemical transformations [13]; (6) study of certain types of carrier
motion in semiconductor theory [14].
Kuznetsov showed in [15] that the integral (6) is also expressible in terms of
the functions Y„ (y, x) or 0 (y, x), but the use of these solutions has proved difficult
n
in practice owing to the absence of detailed tables of Lomriiers functions of two
pure imaginary variables Y (y, x).
n
The present volume gives the values of the functions T (y, x) and Y (y, x)
t 2
to seven significant figures for real values of the variables y, x within the ranges
0^x^y;0^y^ 20.
Some properties of the functions T„ (y, x), 6 (y, x)
n
Lommel's functions of two pure imaginary variables are defined by the
Neumann series (l)-(2) [1, 15]. The graphs of Y (y, x), T (y, x) are illustrated
x 2
in Figs. 1, 2.
The relationships
(?)
(8)
(9)
(10)
follow from the definition of Y„ and 0„. The following relationships should also
be noted:
(U)
Tables of LommeVs Functions xi
FIG. l
ai Tables of LommeVs Functions
FIG. 2
Tables of LommeVs Functions xiii
(12)
We have in particular, when n = 0,
(13)
(14)
If y = x, we obtain by (l)-(2),
(15)
whilst it follows from (11)-(12) that
(16)
(17)
When n = 0, we get from (16)-(17):
(18)
(19)
The same formulae give, with n = 1,
(20)
(21)
If y i= 0, x = 0, we obtain from (2):
(22)
whilst we find from (13)-(14) that
(23)
When y = x = 0, we have
(24)
xiv Tables of LommeVs Functions
It follows from the definition (l)-(2) of the functions Y , 6 that Y , 6 are
2n n 2n 2n
even functions with respect to the variables y and x, whilst Y , 0 +i are even
2n+1 2n
with respect to x and odd with respect to y; for instance,
(25)
Integral forms
The integral forms of the functions T T , 0 and 6 are
l5 2 O X
(26)
(27)
(28)
(29)
where lé (b, x) and lê (b, x) are the integrals
(30)
(31)
and
Asymptotic expansions
The asymptotic expansions of T„, 9 takq different forms, depending on the
n
values of x and y. We introduce the notation a = x¡y, b = i (a + 1/a),
2a = b - 1, 2j? = b 4- 1.
Four cases have to be distinguished in the domain 0 < x < y for large values
of y: (I) a<l; (II) 2<xx>l; (III) 2ax~l; (IV) 2ax<^l; there is a special expansion
for each of these cases.
Tables of LommeVs Functions xv
CASE I. a <^ 1.
We find 9 and 0^ from (2), then T Y from (13)-(14).
0 u 2
We find the values of T Y in the other cases (II-IV) by first using the asymp
u 2
totic expansions of integrals (30)-(31), then using (26)-(27).
CASE II. lax > 1, x > 1.
The asymptotic expansions of integrals (30)-(31) are
(32)
(33)
where
Table 1 below gives the values of the coefficient c = {In — 1) !!/(2w)!! and of
n
[T(- n + i)]"1 for several n ^ 0.
TABLE 1
n
0 1.0 0.56418958 1.0
1 0.5 -0.28209479 0.125
2 0.375 0.42314219 0.070312500
3 0.3125 -1.0578555 0.073242187
4 0.27343750 3.7024941 0.11215210
5 0.24609375 -16.661224 0.22710800
6 0.22558594 91.636730 0.57250142
7 0.20947266 -595.63875 1.7277275
8 0.19638062 4467.2906 6.0740420
9 0.18547058 -37971.970 24.380530
10 0.17619705 360733.72 110.01714