# Book:George Arfken/Mathematical Methods for Physicists/Second Edition

## George Arfken: Mathematical Methods for Physicists (2nd Edition)

Published $\text {1970}$, Academic Press

ISBN 0-12-059832-9.

### Contents

preface to the second edition
preface to the first edition
acknowledgments
introduction

Chapter 1. VECTOR ANALYSIS
1.1 Definitions, elementary approach
1.2 Rotation of coordinates
1.3 Scalar or dot product
1.4 Vector or cross product
1.5 Triple scalar product, triple vector product
1.6 Gradient, $\nabla$
1.7 Divergence, $\nabla \cdot$
1.8 Curl, $\nabla \times$
1.9 Successive applications of $\nabla$
1.10 Vector integration
1.11 Gauss's theorem
1.12 Stokes's theorem
1.13 Potential theory
1.14 Gauss's law, Poisson's equation
1.15 Helmholtz's theorem
References

Chapter 2. COORDINATE SYSTEMS
2.1 Curvilinear coordinates
2.2 Differential vector operations
2.3 Special coordinate systems -- rectangular cartesian coordinates
2.4 Spherical polar coordinates $\tuple {r, \theta, \varphi}$
2.5 Separation of variables
2.6 Circular cylindrical coordinates $\tuple {\rho, \varphi, z}$
2.7 Elliptic cylindrical coordinates $\tuple {u, v, z}$
2.8 Parabolic cylindrical coordinates $\tuple {\xi, \eta, z}$
2.9 Bipolar coordinates $\tuple {\xi, \eta, z}$
2.10 Prolate spheroidal coordinates $\tuple {u, v, \varphi}$
2.11 Oblate spheroidal coordinates $\tuple {u, v, \varphi}$
2.12 Parabolic coordinates $\tuple {\xi, \eta, \varphi}$
2.13 Toroidal coordinates $\tuple {\xi, \eta, \varphi}$
2.14 Bispherical coordinates $\tuple {\xi, \eta, \varphi}$
2.15 Confocal ellipsoidal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
2.16 Confocal coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
2.17 Confocal parabolic coordinates $\tuple {\xi_1, \xi_2, \xi_3}$
References

Chapter 3. TENSOR ANALYSIS
3.1 Introduction, definitions
3.2 Contraction, direct product
3.3 Quotient rule
3.4 Pseudotensors, dual tensors
3.6 Theory of elasticity
3.7 Lorentz covariance of Maxwell's equations
References

Chapter 4. DETERMINANTS, MATRICES, AND GROUP THEORY
4.1 Determinants
4.2 Matrices
4.3 Orthogonal matrices
4.4 Oblique coordinates
4.5 Hermitian matrices, unitary matrices
4.6 Diagonalization of matrices
4.7 Introduction to group theory
4.8 Discrete groups
4.9 Continuous groups
4.10 Generators
4.11 $\map {\mathrm {SU} } 2$, $\map {\mathrm {SU} } 3$ and nuclear particles
4.12 Homogeneous Lorentz Group
References

Chapter 5. INFINITE SERIES
5.1 Fundamental concepts
5.2 Convergence tests
5.3 Alternating series
5.4 Algebra of series
5.5 Series of functions
5.6 Taylor's expansion
5.7 Power series
5.8 Elliptic integrals
5.9 Bernoulli numbers
5.10 Infinite products
5.11 Asymptotic or semiconvergent series
References

Chapter 6. FUNCTIONS OF A COMPLEX VARIABLE I. ANALYTIC PROPERTIES, CONFORMAL MAPPING
6.1 Complex algebra
6.2 Cauchy-Riemann conditions
6.3 Cauchy's integral theorem
6.4 Cauchy's integral formula
6.5 Laurent expansion
6.6 Mapping
6.7 Conformal mapping
6.8 Schwarz-Christoffel transformation
References

Chapter 7. FUNCTIONS OF A COMPLEX VARIABLE II. CALCULUS OF RESIDUES
7.1 Singularities
7.2 Calculus of residues
7.3 Applications of the calculus of residues
7.4 The method of steepest descents

Chapter 8. SECOND-ORDER DIFFERENTIAL EQUATIONS
8.1 Partial differential equations of theoretical physics
8.2 Separation of variables -- ordinary differential equations
8.3 Singular points
8.4 Series solutions -- Frobenius' method
8.5 A second solution
8.6 Nonhomogeneous equation -- Green's function
8.7 Numerical solutions
References

Chapter 9. STURM-LIOUVILLE THEORY -- ORTHOGONAL FUNCTIONS
9.3 Schmidt Orthogonalization
9.4 Completeness of eigenfunctions
References

Chapter 10. THE GAMMA FUNCTION (FACTORIAL FUNCTION)
10.1 Definitions, simple properties
10.2 Digamma and polygamma functions
10.3 Stirling's series
10.4 The beta function
10.5 The incomplete gamma functions and related functions
References

Chapter 11. BESSEL FUNCTIONS
11.1 Bessel functions of the first kind $\map {J_\nu} x$
11.2 Orthogonality
11.3 Neumann functions, Bessel functions of the second kind, $\map {N_\nu} x$
11.4 Hankel functions
11.5 Modified Bessel functions, $\map {I_\nu} x$ and $\map {K_\nu} x$
11.6 Asymptotic expansions
11.7 Spherical Bessel functions
References

Chapter 12. LEGENDRE FUNCTIONS
12.1 Generating function
12.2 Recurrence relations and special properties
12.3 Orthogonality
12.4 Alternate definitions of Legendre polynomials
12.5 Associated Legendre function
12.6 Spherical harmonics
12.7 Angular momentum and ladder operators
12.8 The addition theorem for spherical harmonics
12.9 Integrals of the product of three spherical harmonics
12.10 Legendre functions of the second kind, $\map {Q_n} x$
12.11 Application to spheroidal coordinate systems
12.12 Vector spherical harmonics
References

Chapter 13. SPECIAL FUNCTIONS
13.1 Hermite functions
13.2 Laguerre functions
13.3 Chebyshev (Tschebyscheff) polynomials
13.4 Hypergeometric functions
13.5 Confluent hypergeometric functions
References

Chapter 14. FOURIER SERIES
14.1 General properties
14.2 Advantages, uses of Fourier series
14.3 Applications of Fourier series
14.4 Properties of Fourier series
14.5 Gibbs phenomenon
References

Chapter 15. INTEGRAL TRANSFORMS
15.1 Integral transforms
15.2 Development of the Fourier integral
15.3 Fourier transforms -- inversion theorem
15.4 Fourier transforms of derivatives
15.5 Convolution theorem
15.6 Momentum representation
15.7 Elementary Laplace transforms
15.8 Laplace transform of derivatives
15.9 Other properties
15.10 Convolution or Faltung theorem
15.11 Inverse Laplace transformation
References

Chapter 16. INTEGRAL EQUATIONS
16.1 Introduction
16.2 Integral transforms, generating functions
16.3 Neumann series, separable (degenerate) kernels
16.4 Hilbert-Schmidt theory
16.5 Green's function -- one dimension
16.6 Green's functions -- two and three dimensions
References

Chapter 17. CALCULUS OF VARIATIONS
17.1 One dependent and one independent variable
17.2 Applications of the Euler equation
17.3 Generalizations, several dependent variables
17.4 Several independent variables
17.5 More than one dependent, more than one independent variable
17.6 Lagrangian multipliers
17.7 Variation subject to constraints
17.8 Rayleigh-Ritz variational technique
References

GENERAL REFERENCES

index

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