,Published by
Jenny Stanford Publishing Pte. Ltd.
101 Thomson Road
#06-01, United Square
Singapore 307591
Email:
Web: www.jennystanford.com
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Mathematical Methods of Physics: Problems with Solutions
c 2025 Jenny Stanford Publishing Pte. Ltd.
Copyright
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by any means, electronic or mechanical, including photocopying, recording or
any information storage and retrieval system now known or to be invented, without
written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the
Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In
this case permission to photocopy is not required from the publisher.
ISBN 978-981-5129-21-2 (Hardcover)
ISBN 978-1-003-51360-5 (eBook)
, Contents
Preface to the English Edition ix
Preface to the First Edition xi
1 Linear Operators 1
1.1 Finite Dimensional Space 1
1.2 Functionals and Generalized Functions 4
1.3 Hilbert Space and Completeness 5
1.4 Self-Adjoint Operators 6
1.5 Ket- and Bra- Vectors 8
2 Method of Characteristics 27
2.1 Linear First-Order PDE 27
2.2 Quasilinear Equation 29
2.3 System of Equations 30
3 Second-Order Linear Equations 53
3.1 Canonical Form 53
3.2 Curvilinear Coordinates 55
3.3 Separation of Variables 56
3.4 Fourier Method 57
4 Self-Similarity and Nonlinear Equations 91
4.1 Symmetry of Equations 91
4.2 Nonlinear Equations 92
, vi Contents
5 Special Functions 119
5.1 Singular Points 119
5.2 Hypergeometric Functions 122
5.3 Orthogonal Polynomials 123
6 Asymptotic Methods 159
6.1 Asymptotic Power Series 159
6.2 A Laplace Integral 160
6.3 Method of Stationary Phase 161
6.4 Method of Steepest Descents 161
6.5 The Averaging Method 165
7 Green’s Functions Method 183
7.1 Green’s Functions 183
7.2 Continuous Spectrum 189
7.3 Resolvent 191
8 Integral Equations 223
8.1 Fredholm Equations 223
8.2 Degenerate Kernel 225
8.3 Symmetric Kernel 226
8.4 Inverse Problem for Schrödinger Operator 227
9 Groups and Representations 251
9.1 Groups 251
9.2 Representations 252
10 Continuous Groups 265
10.1 Lie Groups and Algebras 265
10.2 Representations of the Rotation Group 267
11 Group Theory in Physics 291
11.1 Molecular Oscillations 291
11.2 Level Splitting 296
11.3 Selection Rules 297
11.4 Invariant Tensors 299
Jenny Stanford Publishing Pte. Ltd.
101 Thomson Road
#06-01, United Square
Singapore 307591
Email:
Web: www.jennystanford.com
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Mathematical Methods of Physics: Problems with Solutions
c 2025 Jenny Stanford Publishing Pte. Ltd.
Copyright
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by any means, electronic or mechanical, including photocopying, recording or
any information storage and retrieval system now known or to be invented, without
written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the
Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In
this case permission to photocopy is not required from the publisher.
ISBN 978-981-5129-21-2 (Hardcover)
ISBN 978-1-003-51360-5 (eBook)
, Contents
Preface to the English Edition ix
Preface to the First Edition xi
1 Linear Operators 1
1.1 Finite Dimensional Space 1
1.2 Functionals and Generalized Functions 4
1.3 Hilbert Space and Completeness 5
1.4 Self-Adjoint Operators 6
1.5 Ket- and Bra- Vectors 8
2 Method of Characteristics 27
2.1 Linear First-Order PDE 27
2.2 Quasilinear Equation 29
2.3 System of Equations 30
3 Second-Order Linear Equations 53
3.1 Canonical Form 53
3.2 Curvilinear Coordinates 55
3.3 Separation of Variables 56
3.4 Fourier Method 57
4 Self-Similarity and Nonlinear Equations 91
4.1 Symmetry of Equations 91
4.2 Nonlinear Equations 92
, vi Contents
5 Special Functions 119
5.1 Singular Points 119
5.2 Hypergeometric Functions 122
5.3 Orthogonal Polynomials 123
6 Asymptotic Methods 159
6.1 Asymptotic Power Series 159
6.2 A Laplace Integral 160
6.3 Method of Stationary Phase 161
6.4 Method of Steepest Descents 161
6.5 The Averaging Method 165
7 Green’s Functions Method 183
7.1 Green’s Functions 183
7.2 Continuous Spectrum 189
7.3 Resolvent 191
8 Integral Equations 223
8.1 Fredholm Equations 223
8.2 Degenerate Kernel 225
8.3 Symmetric Kernel 226
8.4 Inverse Problem for Schrödinger Operator 227
9 Groups and Representations 251
9.1 Groups 251
9.2 Representations 252
10 Continuous Groups 265
10.1 Lie Groups and Algebras 265
10.2 Representations of the Rotation Group 267
11 Group Theory in Physics 291
11.1 Molecular Oscillations 291
11.2 Level Splitting 296
11.3 Selection Rules 297
11.4 Invariant Tensors 299
