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.
No part of this publication may be reproduced or transmitted in any form or
by any means, electronic or mechanical, including photocopying, recording, or
any information storage and retrieval system, without permission in writing
from the publisher. Details on how to seek permission and further information
about the Publishers permissions policies and our arrangements with organi-
zations such as the Copyright Clearance Center and the Copyright Licensing
Agency, can be found at our website: www.elsevier.com/permissions.
This book and the individual contributions contained in it are protected under
copyright by the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new
research and experience broaden our understanding, changes in research meth-
ods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and
knowledge in evaluating and using any information, methods, compounds,
or experiments described herein. In using such information or methods they
should be mindful of their own safety and the safety of others, including
parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, con-
tributors, or editors, assume any liability for any injury and/or damage to
persons or property as a matter of products liability, negligence or otherwise,
or from any use or operation of any methods, products, instructions, or ideas
contained in the material herein.
For information on all Academic Press publications,
visit our website: www.books.elsevier.com
,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
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.
No part of this publication may be reproduced or transmitted in any form or
by any means, electronic or mechanical, including photocopying, recording, or
any information storage and retrieval system, without permission in writing
from the publisher. Details on how to seek permission and further information
about the Publishers permissions policies and our arrangements with organi-
zations such as the Copyright Clearance Center and the Copyright Licensing
Agency, can be found at our website: www.elsevier.com/permissions.
This book and the individual contributions contained in it are protected under
copyright by the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new
research and experience broaden our understanding, changes in research meth-
ods, professional practices, or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and
knowledge in evaluating and using any information, methods, compounds,
or experiments described herein. In using such information or methods they
should be mindful of their own safety and the safety of others, including
parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, con-
tributors, or editors, assume any liability for any injury and/or damage to
persons or property as a matter of products liability, negligence or otherwise,
or from any use or operation of any methods, products, instructions, or ideas
contained in the material herein.
For information on all Academic Press publications,
visit our website: www.books.elsevier.com
,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
