Solved Problems in Analysis: As Applied to Gamma, Beta, Legendre and Bessel Functions

418 Pages·2013·23.4 MB·English

by  Orin J. Farrell

SOLVED PROBLEMS IN
ANALYSIS
As Applied to Gamma, Beta, Legendre and
Bessel Functions
Orin J. Farrell
Union College
Bertram Ross
New Haven College
Copyright
Copyright © 1963 by Orin J. Farrell and Bertram Ross Copyright © renewed 1991 by Mabel W. Farrell,
James A. Farrell, William M. Farrell, and Bertram Ross All rights reserved.
Bibliographical Note
This Dover edition, first published in 1971 and reissued in 2013, is an unabridged and corrected
republication of the work originally published by the Macmillan Company in 1963 under the title: Solved
Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions.
Library of Congress Cataloging-in-Publication Data
Farrell, Orin J.
[Solved problems]
Solved problems in analysis: as applied to gamma, beta, Legendre and Bessel functions / Orin J. Farrell,
Bertram Ross.
pages cm.
Originally published: New York: Macmillan, 1963, under title Solved problems: gamma and beta
functions, Legendre polynomials, Bessel functions.
Includes bibliographical references and index.
eISBN-13: 978-0-486-78308-6
ISBN-10: 0-486-49390-3
1. Harmonic functions—Problems, exercises, etc. 2. Gamma functions—Problems, exercises, etc. I. Ross,
Bertram. II. Title.
QA405.F3 2013
515’.52—dc23
2013014028
Manufactured in the United States by Courier Corporation
49390301 2013
www.doverpublications.com
PREFACE TO THE FIRST EDITION
This book consists of a selection of problems, each with a solution worked out in
detail, dealing with the properties and applications of the Gamma function and
the Beta function, the Legendre polynomials, and the Bessel functions. For those
problems which involved more than mere choice of a suitable formula and
appropriate use thereof, we have often endeavored to present solutions with
emphasis on the considerations raised by the following questions: How does one
make a start in attacking the problem? What theorems and techniques from
algebra, trigonometry, analytic geometry, calculus, and the theory of functions
appear applicable so as to be likely to effect a solution? How and why does one
proceed from one step to the next? What clues present themselves either in the
statement of the problem or in the facts which develop as the attempt at solution
proceeds? What aspects of the problem must be carefully considered so that the
solution will meet the demands of mathematical rigor?
Such an approach usually leads to solutions that are neither brief nor elegant.
We earnestly hope, however, that the lack of brevity and elegance is
compensated by what may be called a naturalness of procedure combined with a
heuristic presentation that make the solutions relatively easy to follow. We hope
also that the solutions presented will be found stimulating, and that they will
help to develop skill in attacking and solving problems in pure and applied
mathematics.
Cursory examination of this book might give the impression of an occasional
haphazard choice of problem. But no problem was originated or chosen at
random. Selection of problems was made so as to fulfill such purposes as
exposition of suitable techniques of procedure and reasonable coverage of
relevant topics. Often a problem that seems out of place in one of the chapters on
the properties of the functions (Chapters I, III and V), and not closely concerned
with the development of the outstanding properties of a function, will be found
to serve as a useful lemma in one or more later chapters on the applications of
the functions. Indeed, a goodly number of the problems and exercises in
Chapters I, III and V are put to use in the chapters on applications.
References to individual texts or treatises have been used sparingly in the
statements of the problems and in the solutions. However, a modest bibliography
of works typical of those one would find it profitable to consult is included at the
end of the book.
We gratefully make the following acknowledgments: Table III-2 is
reproduced from W. E. Byerly’s Fourier’s Series and Spherical Harmonics with
the permission of Ginn and Company; Tables V-2 through V-27 are printed,
with slight modifications and deletions, from N. W. McLachlan’s Bessel
Functions for Engineers with the permission of Professor N. W. McLachlan and
the Oxford Press; material was used from G. M. Watson’s Theory of Bessel
Functions with the permission of The Cambridge University Press; Tables V-14
through V-21 were reprinted with the permission of The Royal Society and the
American Institute of Electrical Engineers.
We appreciate especially the excellent constructive criticisms and suggestions
made by Dr. Melvin Hausner of New York University.
PREFACE TO THE DOVER EDITION
We have been pleased at the response to this text from students who are studying
applied classical analysis for the first time, and by professors who are not only
looking for ways to motivate but also for ways to bring difficult subject matter
down to an understandable level. In this Dover edition, we have endeavored to
correct errors in the first edition, some of which were discovered by our
students. We also appreciate the very careful reading given by Professor Yoshio
Matsuoka, Kagoshimashi.
CONTENTS
Chapter I
The Gamma Function and the Beta Function
Introduction, Table and Graph
Integral Expressions of Γ(x) Probs. 1–3
Properties of Γ(x) Probs. 4, 10, 12–
21, 31
Specific Evaluations of Γ(x) Probs. 5–9, 11,
22, 23
Infinite Product Expression of Γ(x) Prob. 24
Γ′(1) = Negative of Euler’s Constant Prob. 32
Logarithmic Derivative of Γ(x) for Positive Integers Prob. 33
Integral Expressions of B(x, y) Probs. 25, 27, 28
Properties of B(x, y) Probs. 26, 29, 30
Table of Formulas
Chapter II
Applications of the Gamma Function and the Beta Function
Introduction
Evaluation of Integrals Probs. 1–11, 14–
27, 33, 35
Probs. 12, 13
Infinite Product Expression of π/2 Probs. 28, 29
Evaluation of Certain Geometrical Magnitudes Probs. 30–32, 34,
36–39
Evaluation of Certain Physical Quantities Probs. 36, 40–42
Approximation of n! for Large Integers Probs. 43–47
Two Problems in Probability Probs. 48–49
A Problem in Heat Flow in a Straight Wire Prob. 50
Chapter III
Legendre Polynomials
Introduction
Coefficients in Expansion of a Generating Function Prob. 1
Recurrence Relations Probs. 2, 13–15,
23, 24
Laplace’s Integral Expression of P (x) Prob. 3
n
Determination of Specific Legendre Polynomials Prob. 4–9
Rodrigues’s Expression for P (x) Prob. 10
n
Orthogonality Property and Related Property Probs. 16, 17
Expansion of a Given Function in Legendre Polynomials Probs. 11, 18, 19
|P (x)| 1 for −1 x 1
n
Character and Location of Zeros of P (x) Prob. 12
n
Evaluation of Integrals Involving Legendre Polynomials Prob. 21
Evaluation of P (O) Probs. 20, 26, 28–
2n
36
Evaluation of Derivatives Prob. 25
Tables and Graph
Chapter IV
Applications of Legendre Polynomials
Introduction
Specific Series Expansions Probs. 1–8, 16, 17
Steady-state Heat-flow Temperature Distribution Probs. 9–13
Gravitational Potential of a Circular Lamina Prob. 14
Potential of an Electric Charge Distribution Prob. 15
Infinite Product Expression for π/2 Prob. 18
Application of Gauss’s Mechanical Quadrature Formula Probs. 19–20
with Pertinent Table
Chapter V
Bessel Functions
Introduction
Differentiation Formulas Probs. 1–6, 8–10,
12, 15, 19
Recursion Formulas Probs. 7, 16
Specific Evaluations Probs. 13, 14
Generating Functions Probs. 22, 23
Orthogonality Property and Related Property Probs. 24, 25
Expansion of a Given Function in Bessel Functions Probs. 26, 27
Evaluation of Integrals Involving Bessel Functions Probs. 11, 28–33
Functions of Orders Probs. 17, 18, 20
Alternation of Zeros Prob. 21
Approximations for Small and Large Arguments Probs. 34, 35
Integral Expression of J (x) and of J (x) Probs. 36, 37
p n
Relations to Legendre Polynomials Probs. 38–41
Formulas and Tables
Chapter VI
Applications of Bessel Functions
Introduction
Solutions of Equations Reducible to Bessel’s Equation Probs. 1–6
Specific Expansions in Bessel Functions Probs. 7–10
Problems in Dynamics Probs. 11–17, 21,
22
Flux Distribution in a Nuclear Reactor Prob. 18
Heat-flow Temperature Distribution Probs. 19, 20, 24,
25, 30
Fluid Velocity Imparted by Radially Pulsating Cylinder Prob. 23
Displacement of Vibrating Annular Membrane Prob. 26
Alternating Current Density Prob. 27
Eddy Current Density and Power Loss Probs. 28, 29
Index
I
THE GAMMA FUNCTION AND THE
BETA FUNCTION
INTRODUCTION
The Gamma function was first defined in 1729 by the great Swiss
mathematician Euler. He defined the Gamma function by an infinite product:
If z be taken as the complex variable x + iy, Euler’s product for Γ(z) converges at
every finite z except z = 0, −1, −2, −3, · · ·. The function defined by the product
is analytic at every finite z except for the singular points just mentioned. At each
of the singular points, Γ(z) has a simple pole.
The notation Γ(z) and the name “Gamma function” were first used by
Legendre in 1814.
From Euler’s infinite product for Γ(z) can be derived the formula
This integral formula is convergent only when the real part of z is positive.
Nevertheless this integral formula for Γ(z) often is taken as the starting point for
introductory treatments of the Gamma function. Moreover, the variable z is often
confined to real values x. So shall it be in this book: unless the contrary is
explicitly stated, we shall be concerned in our exercises and problems with the
Gamma function of a real variable only. For positive values of x we shall take
the following as our basic definition of the Gamma function: