Contents v 7 The Fourier Transform and Its Applications 389 7.1 The Fourier Integral Representation 390 7.2 The Fourier Transform 398 7.3 The Fourier Transform Method 411 7.4 The Heat Equation and Gauss's Kernel 420 7.5 A Dirichlet Problem and the Poisson Integral Formula 429 7.6 The Fourier Cosine and Sine Transforms 433 7.7 Problems Involving Semi-Infinite Intervals 440 7.8 Generalized Functions 445 7.9 The Nonhomogeneous Heat Equation 461 7.10 Duhamel's Principle 471 8 The Laplace and Hankel Transforms with Applications 479 8.1 The Laplace Transform 480 8.2 Further Properties of the Laplace Transform 491 8.3 The Laplace Transform Method 502 8.4 The Hankel Transform with Applications 508 9 Finite Difference Numerical Methods 515 9.1 The Finite Difference Method for the Heat Equation 516 9.2 The Finite Difference Method for the Wave Equation 525 9.3 The Finite Difference Method for Laplace's Equation 533 9.4 Iteration Methods for Laplace's Equation 541 10 Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations 546 10.1 The Sampling Theorem 547 10.2 Partial Differential Equations and the Sampling Theorem 555 10.3 The Discrete and Fast Fourier Transforms 559 10.4 The Fourier and Discrete Fourier Transforms 567 11 An Introduction to Quantum Mechanics 573 11.1 Schrodinger's Equation 574 11.2 The Hydrogen Atom 581 11.3 Heisenberg's Uncertainty Principle 590 Supplement on Orthogonal Polynomials 11.4 Hermite and Laguerre Polynomials 597