Based on the author‘s junior-level undergraduate course, this introductory textbook is designed for a course in mathematical physics. Focusing on the physics of oscillations and waves, A Course in Mathematical Methods for Physicists helps students understand the mathematical techniques needed for their future studies in physics. It takes a bottom-up approach that emphasizes physical applications of the mathematics. The book offers: A quick review of mathematical prerequisites, proceeding to applications of differential equations and linear algebra Classroom-tested explanations of complex and Fourier analysis for trigonometric and special functions Coverage of vector analysis and curvilinear coordinates for solving higher dimensional problems Sections on nonlinear dynamics, variational calculus, numerical solutions of differential equations, and Green's functions

Table of Contents:
Prologue, Introduction, What Is Mathematical Physics?, An Overview of the Course, Tips for Students, Acknowledgments, 1 Introduction and Review, 1.1 What Do I Need to Know from Calculus?, 1.1.1 Introduction, 1.1.2 Trigonometric Functions, 1.1.3 Hyperbolic Functions, 1.1.4 Derivatives, 1.1.5 Integrals, 1.1.6 Geometric Series, 1.1.7 Power Series, 1.1.8 Binomial Expansion, 1.2 What I Need from My Intro Physics Class?, 1.3 Technology and Tables, 1.4 Appendix: Dimensional Analysis, Problems, 2 Free Fall and Harmonic Oscillators, 2.1 Free Fall, 2.2 First-Order Differential Equations, 2.2.1 Separable Equations, 2.2.2 Linear First-Order Equations, 2.2.3 Terminal Velocity, 2.3 Simple Harmonic Oscillator, 2.3.1 Mass-Spring Systems, 2.3.2 Simple Pendulum, 2.4 Second-Order Linear Differential Equations, 2.4.1 Constant Coefficient Equations, 2.5 LRC Circuits, 2.5.1 Special Cases, 2.6 Damped Oscillations, 2.7 Forced Systems, 2.7.1 Method of Undetermined Coefficients, 2.7.2 Periodically Forced Oscillations, 2.7.3 Method of Variation of Parameters, 2.7.4 Initial Value Green’s Functions, 2.8 Cauchy–Euler Equations, 2.9 Numerical Solutions of ODEs, 2.9.1 Euler’s Method, 2.9.2 Higher-Order Taylor Methods, 2.9.3 Runge–Kutta Methods, 2.10 Numerical Applications, 2.10.1 Nonlinear Pendulum, 2.10.2 Extreme Sky Diving, 2.10.3 Flight of Sports Balls, 2.10.4 Falling Raindrops, 2.10.5 Two-Body Problem, 2.10.6 Expanding Universe, 2.10.7 Coefficient of Drag, 2.11 Linear Systems, 2.11.1 Coupled Oscillators, 2.11.2 Planar Systems, 2.11.3 Equilibrium Solutions and Nearby Behaviors, 2.11.4 Polar Representation of Spirals, Problems, 3 Linear Algebra, 3.1 Finite Dimensional Vector Spaces, 3.2 Linear Transformations, 3.2.1 Active and Passive Rotations, 3.2.2 Rotation Matrices, 3.2.3 Matrix Representations, 3.2.4 Matrix Inverses and Determinants, 3.2.5 Cramer’s Rule, 3.3 Eigenvalue Problems, 3.3.1 An Introduction to Coupled Systems, 3.3.2 Eigenvalue Problems, 3.3.3 Rotations of Conics, 3.4 Matrix Formulation of Planar Systems, 3.4.1 Solving Constant Coefficient Systems in 2D, 3.4.2 Examples of the Matrix Method, 3.4.3 Planar Systems—Summary, 3.5 Applications, 3.5.1 Mass-Spring Systems, 3.5.2 Circuits, 3.5.3 Mixture Problems, 3.5.4 Chemical Kinetics, 3.5.5 Predator Prey Models, 3.5.6 Love Affairs, 3.5.7 Epidemics, 3.6 Appendix: Diagonalization and Linear Systems, Problems, 4 Nonlinear Dynamics, 4.1 Introduction, 4.2 Logistic Equation, 4.2.1 Riccati Equation, 4.3 Autonomous First-Order Equations, 4.4 Bifurcations for First-Order Equations, 4.5 Nonlinear Pendulum, 4.5.1 Period of the Nonlinear Pendulum, 4.6 Stability of Fixed Points in Nonlinear Systems, 4.7 Nonlinear Population Models, 4.8 Limit Cycles, 4.9 Nonautonomous Nonlinear Systems, 4.10 Exact Solutions Using Elliptic Functions, Problems, 5 Harmonics of Vibrating Strings, 5.1 Harmonics and Vibrations, 5.2 Boundary Value Problems, 5.3 Partial Differential Equations, 5.4 1D Heat Equation, 5.5 1D Wave Equation, 5.6 Introduction to Fourier Series, 5.7 Fourier Trigonometric Series, 5.8 Fourier Series over Other Intervals, 5.8.1 Fourier Series on [a,b], 5.9 Sine and Cosine Series, 5.10 Solution of the Heat Equation, 5.11 Finite Length Strings, 5.12 Gibbs Phenomenon, 5.13 Green’s Functions for 1D Partial Differential Equations, 5.13.1 Heat Equation, 5.13.2 Wave Equation, 5.14 Derivation of Generic 1D Equations, 5.14.1 Derivation of Wave Equation for String, 5.14.2 Derivation of 1D Heat Equation, Problems, 6 Non-Sinusoidal Harmonics and Special Functions, 6.1 Function Spaces, 6.2 Classical Orthogonal Polynomials, 6.3 Fourier–Legendre Series, 6.3.1 Properties of Legendre Polynomials, 6.3.2 Generating Functions: Generating Function for Legendre Polynomials, 6.3.3 Differential Equation for Legendre Polynomials, 6.3.4 Fourier–Legendre Series, 6.4 Gamma Function, 6.5 Fourier–Bessel Series, 6.6 Sturm–Liouville Eigenvalue Problems, 6.6.1 Sturm–Liouville Operators, 6.6.2 Properties of Sturm–Liouville Eigenvalue Problems, 6.6.3 Adjoint Operators, 6.6.4 Lagrange’s and Green’s Identities, 6.6.5 Orthogonality and Reality, 6.6.6 Rayleigh Quotient, 6.6.7 Eigenfunction Expansion Method, 6.7 Nonhomogeneous Boundary Value Problems: Green’s Functions, 6.7.1 Boundary Value Green’s Function, 6.7.2 Properties of Green’s Functions, 6.7.3 Differential Equation for the Green’s Function, 6.7.4 Series Representations of Green’s Functions, 6.7.5 Nonhomogeneous Heat Equation, 6.8 Appendix: Least Squares Approximation, 6.9 Appendix: Fredholm Alternative Theorem, Problems, 7 Complex Representations of Functions, 7.1 Complex Representations of Waves, 7.2 Complex Numbers, 7.3 Complex Valued Functions, 7.3.1 Complex Domain Coloring, 7.4 Complex Differentiation, 7.5 Complex Integration, 7.5.1 Complex Path Integrals, 7.5.2 Cauchy’s Theorem, 7.5.3 Analytic Functions and Cauchy’s Integral Formula, 7.5.4 Laurent Series, 7.5.5 Singularities and Residue Theorem, 7.5.6 Infinite Integrals, 7.5.7 Integration over Multivalued Functions, 7.5.8 Appendix: Jordan’s Lemma, Problems, 8 Transform Techniques in Physics, 8.1 Introduction, 8.1.1 Example 1: Linearized KdV Equation, 8.1.2 Example 2: Free Particle Wave Function, 8.1.3 Transform Schemes, 8.2 Complex Exponential Fourier Series, 8.3 Exponential Fourier Transform, 8.4 Dirac Delta Function, 8.5 Properties of the Fourier Transform, 8.5.1 Fourier Transform Examples, 8.6 Convolution Operation, 8.6.1 Convolution Theorem for Fourier Transforms, 8.6.2 Application to Signal Analysis, 8.6.3 Parseval’s Equality, 8.7 Laplace Transform, 8.7.1 Properties and Examples of Laplace Transforms, 8.8 Applications of Laplace Transforms, 8.8.1 Series Summation Using Laplace Transforms, 8.8.2 Solution of ODEs Using Laplace Transforms, 8.8.3 Step and Impulse Functions, 8.9 Convolution Theorem, 8.10 Inverse Laplace Transform, 8.11 Transforms and Partial Differential Equations, 8.11.1 Fourier Transform and the Heat Equation, 8.11.2 Laplace’s Equation on the Half Plane, 8.11.3 Heat Equation on Infinite Interval, Revisited, 8.11.4 Nonhomogeneous Heat Equation, Problems, 9 Vector Analysis and EM Waves, 9.1 Vector Analysis, 9.1.1 A Review of Vector Products, 9.1.2 Differentiation and Integration of Vectors, 9.1.3 Div, Grad, Curl, 9.1.4 Integral Theorems, 9.1.5 Vector Identities, 9.1.6 Kepler Problem, 9.2 Electromagnetic Waves, 9.2.1 Maxwell’s Equations, 9.2.2 Electromagnetic Wave Equation, 9.2.3 Potential Functions and Helmholtz’s Theorem, 9.3 Curvilinear Coordinates, 9.4 Tensors, Problems, 10 Extrema and Variational Calculus, 10.1 Stationary and Extreme Values of Functions, 10.1.1 Functions of One Variable, 10.1.2 Functions of Several Variables, 10.1.3 Linear Regression, 10.1.4 Lagrange Multipliers and Constraints, 10.2 Calculus of Variations, 10.2.1 Introduction, 10.2.2 Variational Problems, 10.2.3 Euler Equation, 10.2.4 Isoperimetic Problems, 10.3 Hamilton’s Principle, 10.4 Geodesics, Problems, 11 Problems in Higher Dimensions, 11.1 Vibrations of Rectangular Membranes, 11.2 Vibrations of a Kettle Drum, 11.3 Laplace’s Equation in 2D, 11.3.1 Poisson Integral Formula, 11.4 Three-Dimensional Cake Baking, 11.5 Laplace’s Equation and Spherical Symmetry, 11.5.1 Spherical Harmonics, 11.6 Schrödinger Equation in Spherical Coordinates, 11.7 Solution of the 3D Poisson Equation, 11.7.1 Green’s Functions for the 2D Poisson Equation, 11.8 Green’s Functions for Partial Differential Equations, 11.8.1 Introduction, 11.8.2 Laplace’s Equation: .=0, 11.8.3 Homogeneous Time-Dependent Equations, 11.8.4 Inhomogeneous Steady-State Equation, 11.8.5 Inhomogeneous, Time-Dependent Equations, Problems, Appendix Review of Sequences and Infinite Series, A.1 Sequences of Real Numbers, A.2 Convergence of Sequences, A.3 Limit Theorems, A.4 Infinite Series, A.5 Convergence Tests, A.6 Sequences of Functions, A.7 Infinite Series of Functions, A.8 Special Series Expansions, A.9 Order of Sequences and Functions, Problems, Bibliography, Index