This book is a written version of a lecture course that I have conducted over anumber of years at National Tsing Hua University on the subject of mathematical methods for physics In response to student requests, once and again, the main purpose of the book is devoted to motivate students to feel free to study mathematics independently The book is intended to provide advanced undergraduate and beginning graduate students in physical science with the most fundamental tools and important background which they will need in the advanced studies It contains a lot of physical examples appearing in many remarkable reference books, or bursting outin my research career All examples have been carried out step by step as clear as possible Through the numerous illustrations readers may become familiar with the basic outlook of mathematical methods for physics, recognize the skillful techniques in general, and enable to apply or to extend to advanced problems

Preface 3 Contents 5 1 Functions of a Complex Variable 9 11 A Brief Review of Analytic Functions 9 12 Cauchy Residue Theorem and Its Applications 23 13 Poisson's Integral and Mittag-Le_er's Expansion 53 14 Evaluations of Inverse Laplace Transform 58 Exercise 642 Conformal Mapping 69 21 Examples of Conformal Mappings 69 22 Transformation of Harmonic Functions 78 23 Applications to Steady Temperatures 81 24 Applications to Electrostatic Potential 90 25 Schwarz-Christo_el Transformation 101 26 Applications to Fluid Flow 113 Exercise 1243 Elliptic Functions 129 31 Introduction 129 32 Elliptic Integrals 135 33 Parametric Equation of the Ellipse 145 34 Reduction to the Standard Form 157 35 Complex Argument 169 36 _Conformal Mapping 174 37 _Applications 181 Exercise 1964 Tensor Calculus 201 41 Tensor Algebra 201 42 Fundamental Tensor (Metric) 207 43 Parallel Displacement 213 44 Christo_el Symbols 214 45 Covariant Di_erentiation 221 46 Geodesics 228 47 Frenet-Serret Formulas 236 48 Riemann-Christo_el Tensors 241 49 Gravity as a Metric Phenomenon 255 Exercise 2695 Sturm-Liouville Theory 271 51 Adjoint and Hermitian Operators 272 52 Properties of the Hermitian Operators 279 53 Bessel Inequality and Schwarz Inequality 284 54 Green Function 290 55 Gram-Schmidt Orthogonalization 317 Exercise 3216 Gamma Function 323 61 De_nition and Properties of Gamma Functions (z) 323 62 Integral Expression of (z) 328 63 Cauchy and Saalschutz Extension of (z) with Re(z) < 0 332 64 Digamma Functions And Polygamma Functions 333 65 Bernoulli Numbers And Bernoulli Functions 339 66 Euler-Maclaurin Integration Formula 342 67 Beta Function and Incomplete Functions 346 68 Error Functions 353 69 Dirichlet Integral 357 Exercise 3607 Bessel Functions 365 71 Generating Function 365 72 Recurrence Relations 368 73 Integral Expressions of Bessel Function Jn(x) 370 74 Bessel Functions J_(x) with Noninteger _ 371 75 Contour Expression of Bessel Functions 380 76 Orthogonality of Bessel Functions 383 77 The Second Kind Bessel Functions N_(x) 391 78 Hankel Functions H(1;2) _ (x) 394 79 Saddle-Point Method (Steepest Descent) 396 710 Wronskian Formulas 401 711 Modi_ed Bessel Functions 403 712 Spherical Bessel Functions 410 713 Modi_ed Spherical Bessel Functions 419 Exercise 4218 Legendre Functions 425 81 Generating Function 425 82 Recurrence Relations 429 83 Orthogonality 433 84 Rodrigues Formula of Legendre Functions 439 85 Legendre Functions of the Second Kind 444 86 Laplace Integral Representation of Legendre Function 450 87 Associated Legendre Functions 452 88 Spherical Harmonic Functions 462 89 Angular Momentum 468 810 Addition Theorem 474 811 _Integrals of the Product of Three Spherical Harmonic Functions 479 Exercise 4819 Other Special Functions 485 91 Hermite Functions 485 92 Laguerre Functions 502 93 Associated Laguerre Functions 506 94 Chebyshev Polynomials 514 95 Hypergeometric Functions 526 96 Conuent Hypergeometric Functions 535 Exercise 54310 Fourier Series and Fourier Transform 547 101 Fourier Series 547 102 Complex Fourier Series 561 103 Applications to Solving Di_erential Equations 563 104 Fourier Integral 569 105 Properties of Fourier Transform 583 106 Dirac _-Function 601 Exercise 61011 Laplace Transform 615 111 De_nition of Laplace Transform 615 112 Properties of Laplace Transform 618 113 Applications to Special Functions and Di_erential Equations 630 114 Inverse Laplace Transform 647 115 Operator Calculus 656 116 Useful Integrals 662 Exercise 66912 Mellin and Hankel Transform 673 121 De_nition of Integral Transform 673 122 Mellin Transform 678 123 Properties of Mellin Transform 687 124 Hankel Transform 691 125 Properties of Hankel Transform 702 126 Relation Between Hankel and Fourier Transforms 707 127 _Dual Integral Equations 714 128 Finite Hankel Transform 722 Exercise 73813 Integral Equations 741 131 Linear Di_erential Equations And Integral Equations 742 132 Sturm-Liouville Equation into Integral Equation 747 133 Integral Transforms 759 134 Iteration Method 767 135 Separable Kernels 769 136 Eigenvalues and Eigenfunctions 772 137 Variation-Iteration Method 777 138 Two-Dimensional Green Function 782 139 Three-Dimensional Green Function 789 1310Applications to Heat, Wave, and Schrodinger Equations 796 Exercise 80714 Calculus of Variations 813 141 Variational Calculus 814 142 Hamiltonian Principle 821 143 One Dependence, Several Independent Variables 824 144 Several Dependent, Several Independent Variables 827 145 Lagrangian Multipliers 830 146 Variation Subject to Constraints 835 147 Rayleigh-Ritz Method 842 148 Variational Formulation of Eigenfunction Problems 844 149 Eigenfunction Problems by the Ratio Method 850 Exercise 855 Bibliography 859 Index 861Preface 3 Contents 5 1 Functions of a Complex Variable 9 11 A Brief Review of Analytic Functions 9 12 Cauchy Residue Theorem and Its Applications 23 13 Poisson's Integral and Mittag-Le_er's Expansion 53 14 Evaluations of Inverse Laplace Transform 58 Exercise 642 Conformal Mapping 69 21 Examples of Conformal Mappings 69 22 Transformation of Harmonic Functions 78 23 Applications to Steady Temperatures 81 24 Applications to Electrostatic Potential 90 25 Schwarz-Christo_el Transformation 101 26 Applications to Fluid Flow 113 Exercise 1243 Elliptic Functions 129 31 Introduction 129 32 Elliptic Integrals 135 33 Parametric Equation of the Ellipse 145 34 Reduction to the Standard Form 157 35 Complex Argument 169 36 _Conformal Mapping 174 37 _Applications 181 Exercise 1964 Tensor Calculus 201 41 Tensor Algebra 201 42 Fundamental Tensor (Metric) 207 43 Parallel Displacement 213 44 Christo_el Symbols 214 45 Covariant Di_erentiation 221 46 Geodesics 228 47 Frenet-Serret Formulas 236 48 Riemann-Christo_el Tensors 241 49 Gravity as a Metric Phenomenon 255 Exercise 2695 Sturm-Liouville Theory 271 51 Adjoint and Hermitian Operators 272 52 Properties of the Hermitian Operators 279 53 Bessel Inequality and Schwarz Inequality 284 54 Green Function 290 55 Gram-Schmidt Orthogonalization 317 Exercise 3216 Gamma Function 323 61 De_nition and Properties of Gamma Functions9 Other Special Functions 485 91 Hermite Functions 485 92 Laguerre Functions 502 93 Associated Laguerre Functions 506 94 Chebyshev Polynomials 514 95 Hypergeometric Functions 526 96 Conuent Hypergeometric Functions 535 Exercise 54310 Fourier Series and Fourier Transform 547 101 Fourier Series 547 102 Complex Fourier Series 561 103 Applications to Solving Di_erential Equations 563 104 Fourier Integral 569 105 Properties of Fourier Transform 583 106 Dirac _-Function 601 Exercise 61011 Laplace Transform 615 111 De_nition of Laplace Transform 615 112 Properties of Laplace Transform 618 113 Applications to Special Functions and Di_erential Equations 630 114 Inverse Laplace Transform 647 115 Operator Calculus 656 116 Useful Integrals 662 Exercise 66912 Mellin and Hankel Transform 673 121 De_nition of Integral Transform 673 122 Mellin Transform 678 123 Properties of Mellin Transform 687 124 Hankel Transform 691 125 Properties of Hankel Transform 702 126 Relation Between Hankel and Fourier Transforms 707 127 _Dual Integral Equations 714 128 Finite Hankel Transform 722 Exercise 73813 Integral Equations 741 131 Linear Di_erential Equations And Integral Equations 742 132 Sturm-Liouville Equation into Integral Equation 747 133 Integral Transforms 759 134 Iteration Method 767 135 Separable Kernels 769 136 Eigenvalues and Eigenfunctions 772 137 Variation-Iteration Method 777 138 Two-Dimensional Green Function 782 139 Three-Dimensional Green Function 789 1310Applications to Heat, Wave, and Schrodinger Equations 796 Exercise 80714 Calculus of Variations 813 141 Variational Calculus 814 142 Hamiltonian Principle 821 143 One Dependence, Several Independent Variables 824 144 Several Dependent, Several Independent Variables 827 145 Lagrangian Multipliers 830 146 Variation Subject to Constraints 835 147 Rayleigh-Ritz Method 842 148 Variational Formulation of Eigenfunction Problems 844 149 Eigenfunction Problems by the Ratio Method 850 Exercise 855 Bibliography 859 Index 861