Mathematical Methods for Physicists,
7th Edition by Arfken, Weber, Chapter 1-23
SOLUTION MANUAL
,Contents
1 Introduction 1
2 Errata and Revision Status 3
3 Exercise Solutions 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, 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
com- pletely 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 mathe-
matical topics associated with lattice summations and band theory,
• A chapter (32) on Mathieu functions, built using material from two chap-
ters 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 au-
thors at harrishatiqtp.ufl.eduor 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 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
Z ∞
to e−xv
dv. The answer is then correct.
0 1 + v2
Page 610 Exercise 13.1.23 Replace (−t)ν by e −πiνt ν .
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 = 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
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 y n(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 Insert minus sign before P 1 (cos θ).
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 µ
four members of th e
x + ip
first display equation to ¶ ψn(x), and
√
2
change the corresponding member of
µ ¶ the
x − ip
second display equation to
√ ψn (x).
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 The ¡hypergeometric function¢should read
ν 1 ν 3 −2
2F 1
2 + 2 , 2 + 1; ν + 2 ; z .
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
Z cos xt
2 1√ dt.
representation J0(x) = 0 1 − t2
π
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 n(ω), where ω is the transform
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 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 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 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
Page 1015 Exercise 20.7.8 occurrences). 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 Infi te 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 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 + ) .
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
indicating divergence. x
(b) Here the Raabe test P can be written
µ ¶
n+1 1 ln2 (1 + n−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 =− ,
x ln 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.
, THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
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7th Edition by Arfken, Weber, Chapter 1-23
SOLUTION MANUAL
,Contents
1 Introduction 1
2 Errata and Revision Status 3
3 Exercise Solutions 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, 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
com- pletely 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 mathe-
matical topics associated with lattice summations and band theory,
• A chapter (32) on Mathieu functions, built using material from two chap-
ters 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 au-
thors at harrishatiqtp.ufl.eduor 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 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
Z ∞
to e−xv
dv. The answer is then correct.
0 1 + v2
Page 610 Exercise 13.1.23 Replace (−t)ν by e −πiνt ν .
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 = 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
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 y n(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 Insert minus sign before P 1 (cos θ).
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 µ
four members of th e
x + ip
first display equation to ¶ ψn(x), and
√
2
change the corresponding member of
µ ¶ the
x − ip
second display equation to
√ ψn (x).
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 The ¡hypergeometric function¢should read
ν 1 ν 3 −2
2F 1
2 + 2 , 2 + 1; ν + 2 ; z .
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
Z cos xt
2 1√ dt.
representation J0(x) = 0 1 − t2
π
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 n(ω), where ω is the transform
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 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 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 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
Page 1015 Exercise 20.7.8 occurrences). 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 Infi te 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 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 + ) .
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
indicating divergence. x
(b) Here the Raabe test P can be written
µ ¶
n+1 1 ln2 (1 + n−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 =− ,
x ln 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.
, THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
