| Fourier Analysis | |
| Fourier Series | p. 3 |
| Fourier Series of Functions with Periodicity 2¿ | p. 3 |
| Orthogonality of Trigonotric Functions | p. 3 |
| The Fourier Coefficients | p. 5 |
| Expansion of Functions in Fourier Series | p. 6 |
| Convergence of Fourier Series | p. 9 |
| Dirichlet Conditions | p. 9 |
| Fourier Series and Delta Function | p. 10 |
| Fourier Series of Functions of any Period | p. 13 |
| Change of Interval | p. 13 |
| Fourier Series of Even and Odd Functions | p. 21 |
| Fourier Series of Nonperiodic Functions in Limited Range | p. 24 |
| Complex Fourier Series | p. 29 |
| The Method of Jumps | p. 32 |
| Properties of Fourier Series | p. 37 |
| Parseval's Theorem | p. 37 |
| Sums of Reciprocal Powers of Integers | p. 39 |
| Integration of Fourier Series | p. 42 |
| Differentiation of Fourier Series | p. 43 |
| Fourier Series and Differential Equations | p. 45 |
| Differential Equation with Boundary Conditions | p. 45 |
| Periodically Driven Oscillator | p. 49 |
| Exercises | p. 52 |
| Fourier Transforms | p. 61 |
| Fourier Integral as a Limit of a Fourier Series | p. 61 |
| Fourier Cosine and Sine Integrals | p. 65 |
| Fourier Cosine and Sine Transforms | p. 67 |
| Tables of Transforms | p. 72 |
| The Fourier Transform | p. 72 |
| Fourier Transform and Delta Function | p. 79 |
| Orthogonality | p. 79 |
| Fourier Transforms Involving Delta Functions | p. 80 |
| Three-Dimensional Fourier Transform Pair | p. 81 |
| Some Important Transform Pairs | p. 85 |
| Rectangular Pulse Function | p. 85 |
| Gaussian Function | p. 85 |
| Exponentially Decaying Function | p. 87 |
| Properties of Fourier Transform | p. 88 |
| Symmetry Property | p. 88 |
| Linearity, Shifting, Scaling | p. 89 |
| Transform of Derivatives | p. 91 |
| Transform of Integral | p. 92 |
| Parseval's Theorem | p. 92 |
| Convolution | p. 94 |
| Mathematical Operation of Convolution | p. 94 |
| Convolution Theorems | p. 96 |
| Fourier Transform and Differential Equations | p. 99 |
| The Uncertainty of Waves | p. 103 |
| Exercises | p. 105 |
| Sturm-Liouville Theory and Special Functions | |
| Orthogonal Functions and Sturm-Liouville Problems | p. 111 |
| Functions as Vectors in Infinite Dimensional Vector Space | p. 111 |
| Vector Space | p. 111 |
| Inner Product and Orthogonality | p. 113 |
| Orthogonal Functions | p. 116 |
| Generalized Fourier Series | p. 121 |
| Hermitian Operators | p. 123 |
| Adjoint and Self-adjoint (Hermitian) Operators | p. 123 |
| Properties of Hermitian Operators | p. 125 |
| Sturm-Liouville Theory | p. 130 |
| Sturm-Liouville Equations | p. 130 |
| Boundary Conditions of Sturm-Liouville Problems | p. 132 |
| Regular Sturm-Liouville Problems | p. 133 |
| Periodic Sturm-Liouville Problems | p. 141 |
| Singular Sturm-Liouville Problems | p. 142 |
| Green's Function | p. 149 |
| Green's Function and Inhomogeneous Differential Equation | p. 149 |
| Green's Function and Delta Function | p. 150 |
| Exercises | p. 157 |
| Bessel and Legendre Functions | p. 163 |
| Frobenius Method of Differential Equations | p. 164 |
| Power Series Solution of Differential Equation | p. 164 |
| Classifying Singular Points | p. 166 |
| Frobenius Series | p. 167 |
| Bessel Functions | p. 171 |
| Bessel Functions Jn(x) of Integer Order | p. 172 |
| Zeros of the Bessel Functions | p. 174 |
| Gamma Function | p. 175 |
| Bessel Function of Noninteger Order | p. 177 |
| Bessel Function of Negative Order | p. 179 |
| Neumann Functions and Hankel Functions | p. 179 |
| Properties of Bessel Function | p. 182 |
| Recurrence Relations | p. 182 |
| Generating Function of Bessel Functions | p. 185 |
| Integral Representation | p. 186 |
| Bessel Functions as Eigenfunctions of Sturm-Liouville Problems | p. 187 |
| Boundary Conditions of Bessel's Equation | p. 187 |
| Orthogonality of Bessel Functions | p. 188 |
| Normalization of Bessel Functions | p. 189 |
| Other Kinds of Bessel Functions | p. 191 |
| Modified Bessel Functions | p. 191 |
| Spherical Bessel Functions | p. 192 |
| Legendre Functions | p. 196 |
| Series Solution of Legendre Equation | p. 196 |
| Legendre Polynomials | p. 200 |
| Legendre Functions of the Second Kind | p. 202 |
| Properties of Legendre Polynomials | p. 204 |
| Rodrigues' Formula | p. 204 |
| Generating Function of Legendre Polynomials | p. 206 |
| Recurrence Relations | p. 208 |
| Orthogonality and Normalization of Legendre Polynomials | p. 211 |
| Associated Legendre Functions and Spherical Harmonics | p. 212 |
| Associated Legendre Polynomials | p. 212 |
| Orthogonality and Normalization of Associated Legendre Functions | p. 214 |
| Spherical Harmonics | p. 217 |
| Resources on Special Functions | p. 218 |
| Exercises | p. 219 |
| Partial Differential Equations | |
| Partial Differential Equations in Cartesian Coordinates | p. 229 |
| One-Dimensional Wave Equations | p. 230 |
| The Governing Equation of a Vibrating String | p. 230 |
| Separation of Variables | p. 232 |
| Standing Wave | p. 238 |
| Traveling Wave | p. 242 |
| Nonhomogeneous Wave Equations | p. 248 |
| D'Alembert's Solution of Wave Equations | p. 252 |
| Two-Dimensional Wave Equations | p. 261 |
| The Governing Equation of a Vibrating Membrane | p. 261 |
| Vibration of a Rectangular Membrane | p. 262 |
| Three-Dimensional Wave Equations | p. 267 |
| Plane Wave | p. 268 |
| Particle Wave in a Rectangular Box | p. 270 |
| Equation of Heat Conduction | p. 272 |
| One-Dimensional Diffusion Equations | p. 274 |
| Temperature Distributions with Specified Values at the Boundaries | p. 275 |
| Problems Involving Insulated Boundaries | p. 278 |
| Heat Exchange at the Boundary | p. 280 |
| Two-Dimensional Diffusion Equations: Heat Transfer in a Rectangular Plate | p. 284 |
| Laplace's Equations | p. 286 |
| Two-Dimensional Laplace's Equation: Steady-State Temperature in a Rectangular Plate | p. 287 |
| Three-Dimensional Laplace's Equation: Steady-State Temperature in a Rectangular Parallelepiped | p. 289 |
| Helmholtz's Equations | p. 291 |
| Exercises | p. 292 |
| Partial Differential Equations with Curved Boundaries | p. 301 |
| The Laplacian | p. 302 |
| Two-Dimensional Laplace's Equations | p. 304 |
| Laplace's Equation in Polar Coordinates | p. 304 |
| Poisson's Integral Formula | p. 312 |
| Two-Dimensional Helmholtz's Equations in Polar Coordinates | p. 315 |
| Vibration of a Drumhead: Two Dimensional Wave Equation in Polar Coordinates | p. 316 |
| Heat Conduction in a Disk: Two Dimensional Diffusion Equation in Polar Coordinates | p. 322 |
| Laplace's Equations in Cylindrical Coordinates | p. 326 |
| Helmholtz's Equations in Cylindrical Coordinates | p. 331 |
| Three-Dimensional Laplacian in Spherical Coordinates | p. 334 |
| Laplace's Equations in Spherical Coordinates | p. 334 |
| Helmholtz's Equations in Spherical Coordinates | p. 345 |
| Wave Equations in Spherical Coordinates | p. 346 |
| Poisson's Equations | p. 349 |
| Poisson's Equation and Green's Function | p. 351 |
| Green's Function for Boundary Value Problems | p. 355 |
| Exercises | p. 359 |
| Variational Methods | |
| Calculus of Variation | p. 367 |
| The Euler-Lagrange Equation | p. 368 |
| Stationary Value of a Functional | p. 368 |
| Fundamental Theorem of Variational Calculus | p. 370 |
| Variational Notation | p. 372 |
| Special Cases | p. 373 |
| Constrained Variation | p. 377 |
| Solutions to Some Famous Problems | p. 380 |
| The Brachistochrone Problem | p. 380 |
| Isoperimetric Problems | p. 384 |
| The Catenary | p. 386 |
| Minimum Surface of Revolution | p. 391 |
| Fermat's Principle | p. 394 |
| Some Extensions | p. 397 |
| Functionals with Higher Derivatives | p. 397 |
| Several Dependent Variables | p. 399 |
| Several Independent Variables | p. 401 |
| Sturm-Liouville Problems and Variational Principles | p. 403 |
| Variational Formulation of Sturm-Liouville Problems | p. 403 |
| Variational Calculations of Eigenvalues and Eigenfunctions | p. 405 |
| Rayleigh-Ritz Methods for Partial Differential Equations | p. 410 |
| Laplace's Equation | p. 411 |
| Poisson's Equation | p. 415 |
| Helmholtz's Equation | p. 417 |
| Hamilton's Principle | p. 420 |
| Exercises | p. 425 |
| References | p. 431 |
| Index | p. 433******** |
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