Table Of ContentTHE SPECIAL FUNCTIONS
ANDTHEIRAPPROXIMATIONS
Yudell L. Luke
MIDWEST RESEARCH INSTITUTE
KANSAS CITY, MISSOURI
VOLUME II
~
1%9
ACADEMICPRESS New York and London
PREFACE
These volumes are designed to provide scientific workers with a
self-contained and unified development for many of the mathematical
functions which arise in applied problems, as well as the attendant
mathematical theory for their approximations. These functions are
often called the special functions of mathematical physics or more
simply the special functions.
Although the subject of special functions has a long and varied
history, we make no attempt to delve into the many particulars of
Bessel functions, Legendre functions, incomplete gamma functions,
confluent hypergeometric functions, etc., as these data are available
in several sources. We have attempted to give a detailed treatment
of the subject on a broad scale on the basis of which many common
particulars of the above-named functions, as well as of others, can be
derived. Hitherto, much of the material upon which the volumes are
based has been available only in papers scattered throughout the
literature.
The core of special functions is the Gaussian hypergeometricfunction
?!'1 and its confluent forms, the confluent hypergeometric functions
IFI and ifi. The confluent hypergeometric functions slightly modified
are also known as Whittaker functions. The 2Fl includes as special
cases Legendre functions, the incomplete beta function, the complete
elliptic functions of the first and second kinds, and most of the classical
orthogonalpolynomials. Theconfluenthypergeometricfunctions include
as special cases Bessel functions, parabolic cylinder functions, Coulomb
wave functions, and incomplete gamma functions. Numerous properties
of confluent hypergeometric functions flow directly from a knowledge
of the 2Fl' and a basic understanding of the ?!'1 and IFI is sufficient
for the derivation of many characteristics of all the other above-named
functions. A natural generalization of the 2Fl is the generalized hyper-
geometric function, the so-called pFq , which in turn is generalized by
Meijer's G-function. The theory of the pFq and the G-function is
fundamental in the applications, since they contain as special cases
all the commonly used functions of analysis. Further, these functions
vii
viii PREFACE
are the building blocks for many functions which are not members
of the hypergeometric family.
The class of hypergeometric series and functions and G-functions
considered in these volumes are functions of only a single variable.
Known generalizations of such hypergeometric series and functions
include basic hypergeometric series, hypergeometric series in two
or more variables, and G-functions of two or more variables. These
and other possible generalizations have many important applications,
but are not taken up here in view of space requirements. Further, the
theory of approximations for the above-named generalizations analogous
to that for functions of a single variable remains to be developed fully.
Volume I develops the 2FI , IFI , pFq , and the G-functions. Volume
II is mainly concerned with approximations ofthese functions by series
of hypergeometric functions with particular emphasis on expansions
in series of Chebyshev polynomials of the first kind, and with the
approximations of these functions by the ratio of two polynomials.
We call the coefficients in the above Chebyshev polynomial expansions
"Chebyshevcoefficients." Tablesof Chebyshevcoefficientsfor numerous
special functions are given in Volume II. There we also present coeffic-
ients which enter into rational approximations for certain special
functions.
The present work is primarily intended as a reference tool. However,
much of the material can be used as a text for an advanced under-
graduateor graduate course in thespecialfunctions and their approxima-
tions. A two-semester course could be based on the material in Chapters
I-V and selected topics in Chapters VIII-XI. The usual mathematical
topics up to and including the residue calculus of complex variable
theory are a prerequisite. Proofs of many of the key results are given
in detail or sketched. In a few cases the reader is referred to other
sources for proof. Often, results are simply stated without proof as
they follow essentially from previous results. Thus opportunities for
exercises are plentiful.
In a work of this type, special precautions have been taken to ensure
accuracy of all formulas and tables. It is a pleasure to acknowledge
with thanks the valuable assistance rendered by Mrs. Geraldine Coombs
and Miss Rosemary Moran in the preparation of the mathematical
tables. I am particularly grateful to Miss Moran for her assiduous help
in proofreading and in preparing the bibliography and indices. In spite
of all checks imposed to ensure accuracy, it is not reasonable to believe
that the text is error-free. I would appreciate receiving from readers
any criticisms of the material and the identification of any errors.
To acknowledge all sources to which some debt is due is virtually
PREFACE Ix
impossible. The bibliography is extensive. For a critical reading of a
large portion of the manuscript and numerous suggestions leading to
improvement of the text I am indebted to my colleagues Dr. Wyman
Fair, Dr. Jerry Fields, and Dr. Jet Wimp. It has been most rewarding
to have worked with these same colleagues on many technical papers.
Finally, I am pleased to thank the typist, Mrs. Louise Weston, for
her painstaking efforts and devotion to detail in the expert preparation
of the manuscript.
YUDELL L. LUKE
Kansas City, Missouri
February, 1969
AMS 1968 Subject Classifications 3301, 4115, 4117, 4130, 4155, 6505, 6520, 6525
COPYRIGHT © 1969, BY ACADEMIC PREss, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT
WRITTEN PERMISSION FROM THE PUBLISHERS.
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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 68-23498
PRI:',TED IN THE UNITED STATES OF AMERICA
To My Mother
INTRODUCTION
To indicate the extent and scope of the present work, and to identify
its point ofview, a synopsis of the chapters is presented.
Chapter I is devoted to the elements of asymptotic expansions, while
Chapter II takes up the gamma function and related functions. The
./'1 is studied in Chapter III. There the pFqis also introduced because
many results valid for the pF are merely a notational change of results
q
for the 2Fl • This chapter contains two special features. One is a section
on the confluence principle giving conditions so that nontrivial results
known for a pF can be readily extended to deduce results for an rFs ,
q
T ~ p, s ~ q. The other feature is the development of Kummer-type
relations for the logarithmic solutions of the differential equation
satisfied by the 2Fl , quadratic transformation formulas associated with
the logarithmic solutions and evaluation of these solutions for special
values of the argument. The features just noted and other relations
appear in book form for the first time. Chapter IV studies confluent
hypergeometric functions. It is shorter than Chapter III since many
results for the confluent functions readily follow from those for the ./'1 .
The generalized hypergeometric function pF and the G-function
q
are the subject of Chapter V. This is a rather long chapter, and by far
and large, most of the material has hitherto been available only in
research papers. Topics covered include elementary properties, multi-
plication theorems, integral transforms of the G-function, series of
G-functions, expansion theorems, asymptotic expansions of the G-
function, and specialization of these results to the asymptotic expansions
of the pFq• Results on the G-function are most important since each
expression developed becomes a master or key formula from which
many resultsare readilydeduced for the morecommonspecialfunctions.
In the applications it often happens that one might know the name
of a special function, for example, Struve's function (we call this a
"named function"), and would like to know of its properties. It is,
therefore, important to identify Struve's function as a IF2' More
generally, it is convenient to have an index so that a named function
can be identified as a pFqor as a G-function. On the other hand, given
xvii
xviii INTRODUCTION
a pFqor a G-function, we would find it helpful to know whether it is
one of the well-known named special functions. To assist the applied
worker, we have compiled a list of formulas which serve to identify
the pF and G-function notation with the named special functions.
q
These are presented in Chapter VI. There we also give without proof
some key properties of Bessel functions, Lommel functions, and the
incomplete gamma function and related functions.
Asymptotic expansions of the pFqfor large parameters is the subject
of Chapter VII. The material selected for this chapter is taken from
various research papers and is largely governed by results needed in
the development of the approximations studied in Volume II.
Key properties of the classical orthogonal polynomials are set forth
in Chapter VIII. These are given without proof, since almost all the
results are special cases of data given for the ';pI in Chapter III. Topics
pertinent to the approximation of functions are presented. Special
emphasis is given to the evaluation and estimation of coefficients in the
expansion of a given function in series of Chebyshev polynomials of
the first kind. Minimax approximations (that is, best approximations
in the Chebyshev sense) are considered and compared with the corres-
ponding truncated expansion in series of Chebyshev polynomials of the
first kind. The latter are best in the mean square sense. Differential
and integralcharacteristicsofsuch expansions are enumerated. A nesting
procedure is developed to evaluate expeditiously a series of functions
where the functions satisfy a linear finite difference equation. Thus,
expansions in series of orthogonal polynomials can be evaluated in a
manner closely akin to the technique used to sum an ordinary poly-
nomial. The differential and integral properties of expansions in series
of Chebyshev polynomials of the first kind together with the nesting
procedure for their evaluation is most important for the applications,
since one can operate with such expansions directly as one does with
ordinary polynomials without first converting such expansions to an
ordinary polynomial.
The first eight chapters constituteVolumeI. In VolumeII, expansions
of generalized hypergeometric functions and G-functions in series of
functions of the same kind is the subject of Chapter IX. As special
cases we delineate expansions of all the common special functions
previously noted in series of Chebyshev polynomials of the first kind.
These results form the basis for the development of the numerical
values of Chebyshev coefficients which are given in Chapter XVII.
Expansionsfor many ofthe special functions in series of Bessel functions
are also listed in Chapter IX.
Study of rational approximations begins in Chapter X. There the
INTRODUCTION xix
or-method is introduced and used to get polynomial and rational approxi-
mations for the exponential function. For certain values of free para-
meters, it is shown that the rational approximations coalesce with the
approximations which lie on the main diagonal of the Pade table. Pade
approximations to the solution of the first-order Riccati equation and
to the solution of a generalized second-order Riccati equation are
developed. The results for the exponential function are generalized
in Chapter XI to get polynomial and rational approximations for the
pFq and for a certain class of G-functions. When p = 2, q = I, and
one of the numerator parameters is unity, by a special choice of free
parameters we recover well-known Pade approximations. These
approximations which are equivalent to the truncated continued
fractions of Gauss are analyzed in Chapter XIII. Pade approximations
for the incomplete gamma functions are detailed in Chapter XIV.
When p ~ q, the pFq(z) series converges for all z. But when
+
p = q 1, we have convergence only in the unit disk. However, the
function for which the q+1Fq(z) series representation is valid only in
the unit disk iswell definedfor allz, Iarg(l - z) I< 7T. This analytically
continued function is also called q+1Fq(z). The polynomial and rational
approximations developed for the pFq(z) converge for all z when
+ +
p ~ q 1, except that if p = q 1, we must have the restriction
Iarg(l - z) I< 7T. Thus, the approximations in the p = q + I case
converge in a domain where the q+lFq(z) series deverges. Ifp > q + 1,
and the pFq(-z) series does not terminate, then it diverges for all
z -=1= O. In this event, the pFq(-z) series is the asymptotic expansion
of a certain G-function. If p = q + 2, the approximations converge
for Iarg z I< 7T/2 (if P = 0 and one of the numerator parameters is
unity, we have convergence for Iarg z I< 7T), and if p = q+ 3, we
have convergence for z > O. The situation for p ;?: q+ 2 is not fully
understood. Nonetheless, the informationavailable covers avast number
of special functions. We previously remarked that for a special 2Fl
and its confluent forms, the rational approximations are of the Pade
class. Because both the numerator and denominator polynomials of a
Pade approximation satisfy the same three-term recurrence formula,
it is natural to inquire if our rational approximations for the pFqenjoy
a similar property. The answer is in the affirmative, and this and related
topics are taken up in Chapter XII.
Truncated Chebyshev expansions of Chapter IX are best in the mean
square sense, but are not best in the Chebyshev or minimax sense.
For virtually all functions of interest in the applications, there is little
difference. The Chebyshev coefficients for expansions of the pFq and
for a certain class of G-functions are members of the hypergeometric
xx INTRODUCTION
family and asymptotic estimates of these coefficients are available.
Thus a priori evaluation of the effectiveness of such approximations is
known. In contrast, the minimax approximations are not known in
closed form except for a few elementary transcendents. Thus, in general
finite algorithms for the desired coefficients are not available and so
they must be found by an iteration process. Here tabular values of the
function being approximated are required. Acommon way of computing
certain transcendents is by Taylor series. These are in general only
efficient near the point about which the expansion is based. Nonetheless,
+
these expansions have the very desirable feature that the (n l)th
approximation follows from the nth approximation by a simple addition.
The rational approximations described above have a like characteristic.
A striking virtue of the Chebyshev coefficients for the pFq and for a
certain class of G-functions is that they obey a recursion formula of
finite length (Chapter XII), and further, in virtually all instances this
recursion formula when used in the backward direction can produce
numerical values of these coefficients in an efficient manner.
For numerous transcendents characterized by definite integrals,
use of trapezoidal type integration rules provides an efficient scheme
for their computation (Chapter XV).
We have already remarked that expansions in series of Chebyshev
polynomials of the first kind can be used in as natural a manner as one
uses ordinary polynomial expansions. The same is essentially true for
the rational approximations, Our philosophy of approximations is that
they should be as widely applicable in nature as possible. They should
have application not only for evaluation of the functions and com-
putation of zeros of functions, but they also should be useful to get
solutions of differential equations, integral equations, and to invert
transforms. The potential of these approximations is illustrated with
a number of examples in Chapter XVI.
In Chapter XVII, we present tables of Chebyshev coefficients for
many special functions of both hypergeometric and nonhypergeometric
type. For a number of special functions of hypergeometric type, coeffic-
ients in their rational approximations are presented. Some other kinds
of coefficients are also given. The set of Chebyshev coefficients is the
most complete ever assembled. Many of these as well as virtually all
the coefficients in the rational approximations appear here for the
first time.
