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

By Orin J. Farrell and Bertram Ross

Nearly 200 problems, each with a detailed, worked-out solution, deal with the properties and applications of the gamma and beta functions, Legendre polynomials, and Bessel functions. The first two chapters examine gamma and beta functions, including applications to certain geometrical and physical problems such as heat-flow in a straight wire. The following two chapters treat Legendre polynomials, addressing applications to specific series expansions, steady-state heat-flow temperature distribution, gravitational potential of a circular lamina, and application of Gauss's mechanical quadrature formula with pertinent table. The final chapters explore Bessel functions, discussing differentiation formulas, generating functions, relations to Legendre polynomials, and other applications.
This volume constitutes a useful tool for professional engineers and experimental physicists. Students of mathematics, physics, and engineering will particularly benefit from the book's expanded solutions.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 6, 2013
• ISBN: 9780486783086

Book preview

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

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:

As is usually done, we shall extend the domain of the definition of the Gamma function into the realm of negative numbers (exclusive of negative integers) by extrapolation via the characteristic equation

It may be remarked that this function, namely xΓ(x), provides an analytic function whose value at each positive integer n is n!.

The Gamma function itself, as set up by Euler, is such that Γ(n) = (n − 1)! rather than n! when n is a positive integer,

Although the Gamma function was devised by Euler to solve a problem in pure mathematics, here, as elsewhere in mathematics, an invention in pure mathematics has been found useful in applications of mathematics to problems in engineering and the sciences. The Gamma function is particularly useful in certain problems of probability, especially problems that involve factorials of large integers or the incomplete Gamma function

Tables of values of Γ(xx < 2. There is no need to tabulate outside a range whose spread is unity because of the fundamental property Γ(x + 1) = xΓ(xx < 2 is chosen because it is the interval between two successive integers whereon Γ(x) has its lowest values for such an interval, making for economy of tabulation and interpolation.

The Beta function is a function of two arguments. As basic definition for the Beta function B(x, y) we shall take, as is usually done, the definition

The Beta function is related to the Gamma function:

TABLE I-1

The problems worked out in this chapter are mostly exercises dealing with properties and values of the Gamma and Beta functions which can be derived directly from their definitions or which ensue from the identities

At the end of the chapter is a list of the most frequently used formulas.

Figure I-1

The Gamma Function

Problems: Integral Expressions of Γ(x)

I-1. which defines Γ(x) is convergent for every positive x but not convergent for any other real x.

We write first the integral as the sum of two integrals:

where m is any positive number. Let us call the two integrals on the right A and B respectively. We see that A is proper when x 1. On the other hand when x < 1, the second factor of the integrand becomes infinite at t = 0, thus making the integral improper. The first factor e−t does not, of course, cause us any concern in the interval t = 0 to t = m. In fact, since that factor is continuous throughout and becomes unity at t = 0, where the other factor becomes infinite for x < 1, we can conclude by the theory of improper integrals that A is convergent or not. But this last-written integral we know to be convergent when and only when the exponent on t is less than unity. Thus A is convergent when and only when 1 − x < 1, that is, when x is positive.

Integral B is improper for all x simply because the interval is infinite. The problem, then, is to determine the values of x for which it is convergent. In order to do this we first apply to B the formula for integration by parts, namely, ∫ u dv = uv − ∫ v du, taking u = tx−1 and dv = e−tdt:

Now we know by the theory of indeterminate forms (by successive applications of L’Hospital’s Rule) that in the race to infinity et will always win out over any constant power of t for every x. Thus convergence of B now hinges on the convergence of our last-written integral, in which we observe that the exponent on t is less by unity than what it was in B. We keep applying integration by parts to the remaining integral until the exponent on t is nonpositive. (Incidentally, we would not have to do any integrating by parts when x 1.) In any event we finally get for B a finite sum of numbers added to a polynomial in x times an integral of the form

If m be taken sufficiently large, the first factor 1/tp in this last integral is less than unity for all t m, which makes the curve y = 1/tpet lie under the curve y = 1/et for t mis convergent by actual integration. Therefore, our final integral is convergent for every x, which in turn makes B convergent for every x.

The Gamma integral, then, is convergent for those values of x, and only those, for which both A and B are convergent, namely for all positive x:

I-2. Show that

, x > 0 from Eq. (I-1).

Let

For our limits of integration: when t = 0, u = 1; and when t = ∞, u = 0. Then,

I-3.

, x > 0 in Eq. (I-1). Let t = m², then dt = 2m dm. Our limits of integration remain the same. So we have

Problem: Properties of Γ(x)

I-4. Establish the fundamental identity Γ(x + 1) = xΓ(x). *

Before proving the identity directly from the Gamma integral for all positive x we note that this identity is used to define the Gamma function first for −1 < x < 0 by writing it in the form Γ(x) = Γ(x + 1)/x, thence for −2 < x < −1 by the same formula, and so on for all nonintegral negative values of x. It remains, then, to show that Γ(x + 1) = xΓ(x) for every positive x.

Letting x be any positive number, we write the Gamma integral for the argument x + 1:

Next we apply integration by parts, namely ∫u dv = uv − ∫v du, to this latter integral, taking dv = e−tdt and u = tx:

The limit indicated in the first term on the right we know to be zero by the treatment of the indeterminate form ∞/∞ as learned in introductory calculus, using L’Hospital’s Rule (once or twice or several or many times in succession according to the size of x). The second term on the right vanishes, while the third term is none other than xΓ(x). Thus we have

REMARK. It is often found convenient to apply the fundamental identity in one or other of the following forms:

Problems: Specific Evaluations of Γ(x)

I-5. Evaluate Γ(.37). *

We have merely to increase the argument from .37 to 1.37 via the identity Γ(x) = Γ(x + 1)/x so that we can use Table I-1 where Γ(x) is tabulated for 1 < x < 2.

I-6. Evaluate Γ(9/4). *

using the fundamental identity in the form Γ(x) = (x − 1)Γ(x − 1):

I-7. Evaluate Γ(4.6). *

This requires three applications in succession of the identity Γ(x) = (x − l)Γ(x − 1):

I-8. Evaluate Γ(−1.3).

This requires three successive increases of argument by unity via the fundamental identity in the form Γ(x) = Γ(x + 1)/x:

I-9. Show that Γ(l) = 1. *

Putting 1 for x , we have

Problem: Properties of Γ(x)

I-10. If n 2, show that

We start with the result of Prob. I-9: Γ(l) = 1. Then by Prob. I-4 we have Γ(2) = Γ(l + 1) = (l)Γ(l) = 1. Similarly, by continuing to apply the fundamental identity Γ(x + 1) = xΓ(x), we get

At this point we perceive the truth of the formula we have to establish. To prove the formula true we have yet to apply the second stage of the method of proof by mathematical induction. Assume the formula true for an arbitrary integer n 2. Then for the next integer m = n + 1 we have

Thus, the formula also holds for n + 1. But we already know from our work above that it holds for n = 2, n = 3, and n = 4. Consequently, it must hold for the next integer n = 5 and for the next after 5, namely 6, and so on ad infinitum.

REMARKS 1. It is by virtue of the formula just established that the convention of defining and accepting a value for the factorial of zero, namely 0! = 1, came to be adopted. For, if we apply the formula formally with n = 1, we have Γ(1) = (1 − 1)! = 0!. (The exclamation point here may be considered, if you will, as having double significance.) But we know that Γ(l) = 1. So, we agree that zero shall be considered as having a factorial which shall be taken as unity. With this convention for 0! we have

2. Since the Gamma function provides a smooth interpolation function relative to the factorials of the positive integers, it is sometimes used as a means of defining x! when x is nonintegral, i.e., x! = Γ(x + 1). For example (3.6)! = Γ(4.6) ≅ 13.38 by Prob. I-7.

Problem: Specific Evaluation of Γ(x)

I-11. *

for x in . But we do not see any way to evaluate this integral. There is no use trying to integrate by parts because the exponent on t is not a positive integer.

Let us try again. If we change the variable of integration via t = u. This looks a little better. The integrand is not as complicated as before. But we are still baffled when we try to integrate. What to do? We begin at this point to suspect that we may have to resort to some indirect . But what? The following scheme appears to be without motivation. Indeed, its discoverer was surely a person of great mathematical ingenuity.

twice: once with x as the variable of integration, then with y as the variable of integration. Then we multiply the results:

Now, although the right side is the product of two integrals, its appearance suggests an iterated integral. Indeed, in this instance we may actually write

because the integral in y yields a mere constant to carry over into the integral in x. May we now equate the iterated integral to a double integral? Yes, we may:

where Q denotes the entire first quadrant of the cartesian xy-plane. This improper double integral over the entire first quadrant is, of course,

where R denotes the rectangular region 0 ≤ x ≤ M, 0 ≤ y ≤ N, and is, therefore equivalent to

Now, our double integral may likewise be equated to an iterated integral in polar coordinates:

which in turn (by the same argument as used before with xy-co-ordinates) may be expressed as a product of two integrals (since the integral in r yields only a constant value independent of θ). We now have

making

Motivation for the ingenious scheme of evaluation is now apparent. It was the presence of the factor r in the iterated polar-coordinates integral that made the integration possible, and that is what suggested the original multiplication whereby we got from e−x² to e−(x²+y²) = e−r²

Problems: Properties of Γ(x)

I-12. Show that if n be a positive integer, then

can be written (2n + l)/2. If we recall the property of Eq. (I-4.3), namely Γ(x) = (x − 1)Γ(x − 1) and if we take x = (2n + 1)/2, then we have

that is,

The process of decreasing the argument by unity is repeated for Γ[(2n - l)/2]:

So far we have

, for instance, we can write

. This 3 corresponds to the n .

we have to continue decreasing the argument by unity n times in