Instructor’s Manual: Mathematical Methods for Physicists

524 Pages·2014·2.17 MB·English

by  George B. Arfken

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
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Contents
1 Introduction 1
2 ErrataandRevisionStatus 3
3 ExerciseSolutions 7
1. Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . 7
2. Determinants and Matrices . . . . . . . . . . . . . . . . . . . . 27
3. Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4. Tensors and Differential Forms . . . . . . . . . . . . . . . . . . 58
5. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6. Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . 81
7. Ordinary Differential Equations . . . . . . . . . . . . . . . . . 90
8. Sturm-Liouville Theory . . . . . . . . . . . . . . . . . . . . . . 106
9. Partial Differential Equations . . . . . . . . . . . . . . . . . . 111
10. Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 118
11. Complex Variable Theory . . . . . . . . . . . . . . . . . . . . 122
12. Further Topics in Analysis . . . . . . . . . . . . . . . . . . . . 155
13. Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 166
14. Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 192
15. Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . 231
16. Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 256
17. Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
18. More Special Functions . . . . . . . . . . . . . . . . . . . . . . 286
19. Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
20. Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . 332
21. Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . . 364
22. Calculus of Variations. . . . . . . . . . . . . . . . . . . . . . . 373
23. Probability and Statistics . . . . . . . . . . . . . . . . . . . . . 387
4 Correlation,ExercisePlacement 398
5 UnusedSixthEditionExercises 425
iv
Chapter 1
Introduction
TheseventheditionofMathematical Methods for Physicistsisasubstantialand
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
notinpreviouseditions,andtherehasbeenawide-spreadreorganizationofthe
previouslyexistingexercisestooptimizetheirplacementrelativetothematerial
inthetext. Sincemanyinstructorswhohaveusedpreviouseditionsofthistext
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,wheretheproblemsintheneweditioncamefrom. Wehaveincluded
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 com-
pletelyapparentfromtheproblemstatement. Whilemanyoftheproblemsfrom
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
websitearethreechaptersthatwerenotincludedintheprintedtextbutwhich
may be important to some instructors. These include
• A new chapter (designated 31) on Periodic Systems, dealing with mathe-
matical topics associated with lattice summations and band theory,
• A chapter (32) on Mathieu functions, built using material from two chap-
tersinthesixthedition,butexpandedintoasinglecoherentpresentation,
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
placesintheseventheditiontext. ThisalternateChapter1containsnomaterial
notalreadyintheseventheditionbutitssubjectmatterhasbeenpackagedinto
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.
BecausethisInstructor’sManualexistsonlyon-line,thereisanopportunity
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 au-
thorsatharris(cid:104)at(cid:105)qtp.ufl.eduortothepublisher. Ifuserschoosetoforward
additionalmaterialsthatareofgeneralusetoinstructorswhoareteachingfrom
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 assis-
tance 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
(cid:90)
∞ e−xv
to dv. The answer is then correct.
1+v2
0
Page 610 Exercise 13.1.23 Replace (−t)ν by e−πiνtν.
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) C can be determined by requiring consistency
1
with the recurrence formula zΓ(z)=Γ(z+1).
Consistency with the duplication formula then
determines C .
2
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 =ω2/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
a and b (denominator and integration
mn mn
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 n (x) by y (x).
n n
Page 723 Exercise 15.1.12 The last formula of the answer should read
P (0)/(2s+2)=(−1)s(2s−1)!!/(2s+2)!!.
2s
Page 754 Exercise 15.4.10 Insert minus sign before P1(cosθ).
n
√
Page 877 Exercise 18.1.6 In both (a) and (b), change 2π to 2π.
Page 888 Exercise 18.2.7 Change the second of the four members of the
(cid:181) (cid:182)
x+ip
first display equation to √ ψ (x), and
n
2
change the corresponding member of the
(cid:181) (cid:182)
x−ip
second display equation to √ ψ (x).
n
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 T term (if
0
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 The hypergeometric function should read
(cid:161) (cid:162)
F ν + 1,ν +1;ν+ 3;z−2 .
2 1 2 2 2 2
Page 916 Exercise 18.5.10 Change (n− 1)! to Γ(n+ 1).
2 2
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 k2; In the
integrand of the hint, change k to k2.
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) This exercise would have been easier if the
book had mentioned the integral
(cid:90)
2 1 cosxt
representation J (x)= √ dt.
0 π 1−t2
0
Page 978 Exercise 20.2.10(b) Change the argument of the square root
to x2−a2.
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)1/2inj (ω), where ω is the transform
n
variable. The text assumes it to be kr.
Page 980 Exercise 20.2.16 Change d3x to d3r and remove the limits
from the first integral (it is assumed to be
over all space).
Page 980 Eq. (20.54) Replace dk by d3k (occurs three times)
Page 997 Exercise 20.4.10 This exercise assumes that the units and
scaling of the momentum wave function
correspond to the formula
(cid:90)
1
ϕ(p)= ψ(r)e−ir·p/(cid:126)d3r.
(2π(cid:126))3/2
Page 1007 Exercise 20.6.1 The second and third orthogonality equa-
tions 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 (cid:54)=(0 or N);
−N/2, p(cid:54)=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
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
(cid:80)
1.1.1. (a) If u <A/np the integral test shows u converges for p>1.
n n n
(cid:80)
(b) If u >A/n, u diverges because the harmonic series diverges.
n n n
1.1.2. This is valid because a multiplicative constant does not affect the conver-
gence or divergence of a series.
(n+1)ln(1+n−1)
1.1.3. (a) The Raabe test P can be written 1+ .
lnn
This expression approaches 1 in the limit of large n. But, applying the
Cauchy integral test,
(cid:90)
dx
=lnlnx,
xlnx
indicating divergence.
(b) Here the Raabe test P can be written
(cid:181) (cid:182)
n+1 1 ln2(1+n−1)
1+ ln 1+ + ,
lnn n ln2n
which also approaches 1 as a large-n limit. But the Cauchy integral test
yields
(cid:90)
dx 1
=− ,
xln2x lnx
indicating convergence.
1.1.4. Convergent for a −b >1. Divergent for a −b ≤1.
1 1 1 1
1.1.5. (a) Divergent, comparison with harmonic series.
7