Mathematical Methods for Physicists 7th Ed Arfken Solutions Manual PDF

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Instructor’s Manual MATHEMATICAL METHODS FOR PHYSICISTS A Comprehensive Guide SEVENTH EDITION George B. Arfken Miami University Oxford, OH Hans J. Weber University of Virginia Charlottesville, VA Frank E. Harris University of Utah, Salt Lake City, UT; University of Florida, Gainesville, FL AMSTERDAM...

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Instructor’s Manual

MATHEMATICAL METHODS FOR PHYSICISTS A Comprehensive Guide SEVENTH EDITION George B. Arfken Miami University Oxford, OH Hans J. Weber University of Virginia Charlottesville, VA Frank E. Harris University of Utah, Salt Lake City, UT; University of Florida, Gainesville, FL

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK c 2013 Elsevier Inc. All rights reserved.

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Contents 1 Introduction

1

2 Errata and Revision Status

3

3 Exercise Solutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Mathematical Preliminaries . . Determinants and Matrices . . . Vector Analysis . . . . . . . . . Tensors and Differential Forms . Vector Spaces . . . . . . . . . . Eigenvalue Problems . . . . . . Ordinary Differential Equations Sturm-Liouville Theory . . . . . Partial Differential Equations . Green’s Functions . . . . . . . . Complex Variable Theory . . . Further Topics in Analysis . . . Gamma Function . . . . . . . . Bessel Functions . . . . . . . . . Legendre Functions . . . . . . . Angular Momentum . . . . . . . Group Theory . . . . . . . . . . More Special Functions . . . . . Fourier Series . . . . . . . . . . Integral Transforms . . . . . . . Integral Equations . . . . . . . . Calculus of Variations . . . . . . Probability and Statistics . . . .

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7 7 27 34 58 66 81 90 106 111 118 122 155 166 192 231 256 268 286 323 332 364 373 387

4 Correlation, Exercise Placement

398

5 Unused Sixth Edition Exercises

425

iv

Chapter 1

Introduction The seventh edition of Mathematical Methods for Physicists is a substantial and detailed revision of its predecessor. The changes extend not only to the topics and their presentation, but also to the exercises that are an important part of the student experience. The new edition contains 271 exercises that were not in previous editions, and there has been a wide-spread reorganization of the previously existing exercises to optimize their placement relative to the material in the text. Since many instructors who have used previous editions of this text have favorite problems they wish to continue to use, we are providing detailed tables showing where the old problems can be found in the new edition, and conversely, where the problems in the new edition came from. We have included the full text of every problem from the sixth edition that was not used in the new seventh edition. Many of these unused exercises are excellent but had to be left out to keep the book within its size limit. Some may be useful as test questions or additional study material. Complete methods of solution have been provided for all the problems that are new to this seventh edition. This feature is useful to teachers who want to determine, at a glance, features of the various exercises that may not be completely apparent from the problem statement. While many of the problems from the earlier editions had full solutions, some did not, and we were unfortunately not able to undertake the gargantuan task of generating full solutions to nearly 1400 problems. Not part of this Instructor’s Manual but available from Elsevier’s on-line web site are three chapters that were not included in the printed text but which may be important to some instructors. These include • A new chapter (designated 31) on Periodic Systems, dealing with mathematical topics associated with lattice summations and band theory, • A chapter (32) on Mathieu functions, built using material from two chapters in the sixth edition, but expanded into a single coherent presentation, and

1

CHAPTER 1. INTRODUCTION

2

• A chapter (33) on Chaos, modeled after Chapter 18 of the sixth edition but carefully edited. In addition, also on-line but external to this Manual, is a chapter (designated 1) on Infinite Series that was built by collection of suitable topics from various places in the seventh edition text. This alternate Chapter 1 contains no material not already in the seventh edition but its subject matter has been packaged into a separate unit to meet the demands of instructors who wish to begin their course with a detailed study of Infinite Series in place of the new Mathematical Preliminaries chapter. Because this Instructor’s Manual exists only on-line, there is an opportunity for its continuing updating and improvement, and for communication, through it, of errors in the text that will surely come to light as the book is used. The authors invite users of the text to call attention to errors or ambiguities, and it is intended that corrections be listed in the chapter of this Manual entitled Errata and Revision Status. Errata and comments may be directed to the authors at harrishatiqtp.ufl.edu or to the publisher. If users choose to forward additional materials that are of general use to instructors who are teaching from the text, they will be considered for inclusion when this Manual is updated. Preparation of this Instructor’s Manual has been greatly facilitated by the efforts of personnel at Elsevier. We particularly want to acknowledge the assistance of our Editorial Project Manager, Kathryn Morrissey, whose attention to this project has been extremely valuable and is much appreciated. It is our hope that this Instructor’s Manual will have value to those who teach from Mathematical Methods for Physicists and thereby to their students.

Chapter 2

Errata and Revision Status Last changed: 06 April 2012

Errata and Comments re Seventh Edition text Page 522

Exercise 11.7.12(a)

This is not a principal-value integral.

Page 535

Figure 11.26

The two arrowheads in the lower part of the circular arc should be reversed in direction.

Page 539

Exercise 11.8.9

The answer is incorrect; it should be π/2.

Page 585

Exercise 12.6.7

Change the integral for which a series is sought Z ∞ −xv e dv. The answer is then correct. to 1 + v2 0

Page 610

Exercise 13.1.23

Replace (−t)ν by e−πiν tν .

Page 615

Exercise 13.2.6

In the Hint, change Eq. (13.35) to Eq. (13.44).

Page 618

Eq. (13.51)

Change l.h.s. to B(p + 1, q + 1).

Page 624

After Eq. (13.58)

C1 can be determined by requiring consistency with the recurrence formula zΓ(z) = Γ(z + 1). Consistency with the duplication formula then determines C2 .

Page 625

Exercise 13.4.3

Replace “(see Fig. 3.4)” by “and that of the recurrence formula”.

Page 660

Exercise 14.1.25

Note that α2 = ω 2 /c2 , where ω is the angular frequency, and that the height of the cavity is l. 3

CHAPTER 2. ERRATA AND REVISION STATUS

4

Page 665

Exercise 14.2.4

Change Eq. (11.49) to Eq. (14.44).

Page 686

Exercise 14.5.5

In part (b), change l to h in the formulas for amn and bmn (denominator and integration limit).

Page 687

Exercise 14.5.14

The index n is assumed to be an integer.

Page 695

Exercise 14.6.3

The index n is assumed to be an integer.

Page 696

Exercise 14.6.7(b)

Change N to Y (two occurrences).

Page 709

Exercise 14.7.3

In the summation preceded by the cosine function, change (2z)2s to (2z)2s+1 .

Page 710

Exercise 14.7.7

Replace nn (x) by yn (x).

Page 723

Exercise 15.1.12

The last formula of the answer should read P2s (0)/(2s + 2) = (−1)s (2s − 1)!!/(2s + 2)!!.

Page 754

Exercise 15.4.10

Page 877

Exercise 18.1.6

Page 888

Exercise 18.2.7

Change the second of theµfour members of the ¶ x + ip √ first display equation to ψn (x), and 2 change the corresponding member of ¶ the µ x − ip √ ψn (x). second display equation to 2

Page 888

Exercise 18.2.8

Change x + ip to x − ip.

Page 909

Exercise 18.4.14

All instances of x should be primed.

Page 910

Exercise 18.4.24

The text does not state that the T0 term (if present) has an additional factor 1/2.

Page 911

Exercise 18.4.26(b)

The ratio approaches (πs)−1/2 , not (πs)−1 .

Page 915

Exercise 18.5.5

Page 916

Exercise 18.5.10

The hypergeometric function should read ¡ν ¢ 1 ν 3 −2 . 2 F1 2 + 2 , 2 + 1; ν + 2 ; z

Page 916

Exercise 18.5.12

Here n must be an integer.

Page 917

Eq. (18.142)

In the last term change Γ(−c) to Γ(2 − c).

Page 921

Exercise 18.6.9

Change b to c (two occurrences).

Page 931

Exercise 18.8.3

The arguments of K and E are m.

Page 932

Exercise 18.8.6

All arguments of K and E are k 2 ; In the integrand of the hint, change k to k 2 .

Insert minus sign before P 1n (cos θ). √ In both (a) and (b), change 2π to 2π.

Change (n − 12 )! to Γ(n + 12 ).

CHAPTER 2. ERRATA AND REVISION STATUS

5

Page 978

Exercise 20.2.9

The formula as given assumes that Γ > 0.

Page 978

Exercise 20.2.10(a)

Page 978

Exercise 20.2.10(b)

This exercise would have been easier if the book had mentioned the integral Z 2 1 cos xt √ dt. representation J0 (x) = π 0 1 − t2

Page 978

Exercise 20.2.11

The l.h.s. quantities are the transforms of their r.h.s. counterparts, but the r.h.s. quantities are (−1)n times the transforms of the l.h.s. expressions.

Page 978

Exercise 20.2.12

The properly scaled transform of f (µ) is (2/π)1/2 in jn (ω), where ω is the transform variable. The text assumes it to be kr.

Page 980

Exercise 20.2.16

Change d3 x to d3 r and remove the limits from the first integral (it is assumed to be over all space).

Page 980

Eq. (20.54)

Replace dk by d3 k (occurs three times)

Page 997

Exercise 20.4.10

This exercise assumes that the units and scaling of the momentum wave function correspond to theZformula 1 ψ(r) e−ir·p/~ d3 r . ϕ(p) = (2π~)3/2

Page 1007

Exercise 20.6.1

The second and third orthogonality equations are incorrect. The right-hand side of the second equation should read: N , p = q = (0 or N/2); N/2, (p + q = N ) or p = q but not both; 0, otherwise. The right-hand side of the third equation should read: N/2, p = q and p + q 6= (0 or N ); −N/2, p 6= q and p + q = N ; 0, otherwise.

Page 1007

Exercise 20.6.2

The exponentials should be e2πipk/N and e−2πipk/N .

Page 1014

Exercise 20.7.2

This exercise is ill-defined. Disregard it.

Page 1015

Exercise 20.7.6

Replace (ν − 1)! by Γ(ν) (two occurrences).

Page 1015

Exercise 20.7.8

Change M (a, c; x) to M (a, c, x) (two

Change the argument of the square root to x2 − a2 .

CHAPTER 2. ERRATA AND REVISION STATUS

6

occurrences). Page 1028

Table 20.2

Most of the references to equation numbers did not get updated from the 6th edition. The column of references should, in its entirety, read: (20.126), (20.147), (20.148), Exercise 20.9.1, (20.156), (20.157), (20.166), (20.174), (20.184), (20.186), (20.203).

Page 1034

Exercise 20.8.34

Note that u(t − k) is the unit step function.

Page 1159

Exercise 23.5.5

This problem should have identified m as the mean value and M as the “random variable” describing individual student scores.

Corrections and Additions to Exercise Solutions None as of now.

Chapter 3

Exercise Solutions 1.

Mathematical Preliminaries

1.1

Infinite Series

P 1.1.1. (a) If un < A/np the integral test shows n un converges for p > 1. P (b) If un > A/n, n un diverges because the harmonic series diverges.

1.1.2. This is valid because a multiplicative constant does not affect the convergence or divergence of a series. 1.1.3. (a) The Raabe test P can be written 1 +

(n + 1) ln(1 + n−1 ) . ln n

This expression approaches 1 in the limit of large n. But, applying the Cauchy integral test, Z dx = ln ln x, x ln x indicating divergence.

(b) Here the Raabe test P can be written ¶ µ n+1 ln2 (1 + n−1 ) 1 1+ , + ln 1 + ln n n ln2 n which also approaches 1 as a large-n limit. But the Cauchy integral test yields Z dx 1 , =− 2 ln x x ln x indicating convergence. 1.1.4. Convergent for a1 − b1 > 1. Divergent for a1 − b1 ≤ 1. 1.1.5.

(a) Divergent, comparison with harmonic series. 7

8

CHAPTER 3. EXERCISE SOLUTIONS (b) Divergent, by Cauchy ratio test. (c) Convergent, comparison with ζ(2). (d) Divergent, comparison with (n + 1)−1 . (e) Divergent, comparison with 12 (n + 1) 1.1.6.

−1

or by Maclaurin integral test.

(a) Convergent, comparison with ζ(2). (b) Divergent, by Maclaurin integral test. (c) Convergent, by Cauchy ratio test. µ ¶ 1 1 (d) Divergent, by ln 1 + ∼ . n n (e) Divergent, majorant is 1/(n ln n).

1.1.7. The solution is given in the text. 1.1.8. The solution is given in the text. 1.1.10. In the limit of large n, un+1 /un = 1 +

1 + O(n−2 ). n

Applying Gauss’ test, this indicates divergence. 1.1.11. Let sn be the absolute value of the nth term of the series. (a) Because ln n increases less rapidly than n, sn+1 < sn and limn→∞ sn = 0. Therefore this series converges. Because the sn are larger than corresponding terms of the harmonic series, this series is not absolutely convergent. (b) Regarding this series as a new series with terms formed by combining adjacent terms of the same sign in the original series, we have an alternating series of decreasing terms that approach zero as a limit, i.e., 1 1 1 1 + > + , 2n + 1 2n + 2 2n + 3 2n + 4 this series converges. With all signs positive, this series is the harmonic series, so it is not aboslutely convergent. (c) Combining adjacent terms of the same sign, the terms of the new series satisfy µ ¶ µ ¶ µ ¶ µ ¶ 1 1 1 1 1 1 1 1 1 > + >2 , 3 > + + >3 , etc. 2 2 2 3 3 4 4 5 6 6 The general form of these relations is n2

2n 2 > sn > . −n+2 n+1

9

CHAPTER 3. EXERCISE SOLUTIONS

An upper limit to the left-hand side member of this inequality is 2/(n−1). We therefore see that the terms of the new series are decreasing, with limit zero, so the original series converges. With all signs positive, the original series becomes the harmonic series, and is therefore not absolutely convergent. 1.1.12. The solution is given in the text. 1.1.13. Form the nth term of ζ(2)−c1 α1 −c2 α2 and choose c1 and c2 so that when placed over the common denominator n2 (n + 1)(n + 2) the numerator will be independent of n. The values of the ci satisfying this condition are c1 = c2 = 1, and our resulting expansion is ζ(2) = α1 + α2 +

∞

∞ X

5 X 2 2 = + . 2 2 n (n + 1( n + 2) 4 n=1 n (n + 1( n + 2) n=1

Keeping terms through n = 10, this formula yields ζ(2) ≈ 1.6445; to this precision the exact value is ζ(2) = 1.6449. 1.1.14. Make the observation that ∞ X

∞

X 1 1 + = ζ(3) 3 (2n + 1) (2n)3 n=0 n=1 and that the second term on the left-hand side is ζ(3)/8). Our summation therefore has the value 7ζ(3)/8. P∞ 1.1.15. (a) Write ζ(n) − 1 as p=2 p−n , so our summation is ∞ X ∞ ∞ X ∞ X X 1 1 = . n n p p n=2 p=2 p=2 n=2

The summation over n is a geometric series which evaluates to 1 p−2 = 2 . 1 − p−1 p −p Summing now over p, we get ∞ X

∞

X 1 1 = = α1 = 1 . p(p − 1) p=1 p(p + 1) p=2

(b) Proceed in a fashion similar to part (a), but now the geometric series has sum 1/(p2 + p), and the sum over p is now lacking the initial term of α1 , so ∞ X 1 1 1 = α1 − = . p(p + 1) (1)(2) 2 p=2

10

CHAPTER 3. EXERCISE SOLUTIONS 1.1.16. (a) Write ζ(3) = 1 +

=1+

¸ ∞ ∞ ∞ · X X X 1 1 1 1 1 ′ + − + α = 1 + − 2 3 3 2 − 1) n (n − 1)n(n + 1) n n(n 4 n=2 n=2 n=2 ∞

1 1 X − . 4 n=2 n3 (n2 − 1)

(b) Now use α2′ and α4′ =

∞ X

1 1 = : 2 − 1)(n2 − 4) n(n 9...