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FullNotes - MATH 529 – Mathematical Methods for Physical Sciences II Christoph Kirsch Contents 0 Overview of the lecture 3 0.1 Partial

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Unformatted text preview: MATH 529 – Mathematical Methods for Physical Sciences II Christoph Kirsch April 27, 2011 Contents 0 Overview of the lecture 3 0.1 Partial differential equations . . . . . . . . . . . . . . . . . . . 4 0.2 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 Review of Fourier analysis 8 1.1 Fourier transform: definition . . . . . . . . . . . . . . . . . . . 8 1.2 Fourier transform: properties . . . . . . . . . . . . . . . . . . 8 1.3 Periodic functions: Fourier series . . . . . . . . . . . . . . . . 9 1.4 Application: solution of ODEs . . . . . . . . . . . . . . . . . . 10 12 Partial Differential Equations (PDEs) 12 12.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 12 12.2 1D Wave Equation: Vibrating String . . . . . . . . . . . . . . 16 12.3 1D Wave Equation: Separation of Variables, Use of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 12.4 D’Alembert’s Solution of the Wave Equation. Characteristics. 26 12.5 Heat Equation: Solution by Fourier Series . . . . . . . . . . . 31 12.6 1D Heat Equation: Solution by Fourier Integrals and Transforms 38 12.7 2D Wave Equation: Vibrating Membrane . . . . . . . . . . . . 46 12.8 Rectangular Membrane. Double Fourier Series . . . . . . . . . 48 12.9 Laplacian in Polar Coordinates. Circular Membrane. Fourier- Bessel Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1 12.10Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.11Solution of PDEs by Laplace Transforms . . . . . . . . . . . . 72 12.12Application: hydrogen-like atomic orbitals . . . . . . . . . . . 73 12.13Application: Unbounded domains and artificial boundaries. Dirichlet-to-Neumann map. . . . . . . . . . . . . . . . . . . . 76 12.14Review of Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . 79 12.14.1Basic Concepts (12.1) . . . . . . . . . . . . . . . . . . . 79 12.14.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . 80 12.14.3Separation of Variables (12.3, 12.5, 12.8, 12.9, 12.10) . 80 12.14.4Method of Characteristics (12.4) . . . . . . . . . . . . . 88 12.14.5Fourier Integrals (12.6) . . . . . . . . . . . . . . . . . . 90 13 Complex Numbers and Functions 92 13.1 Complex Numbers. Complex Plane. . . . . . . . . . . . . . . . 92 13.2 Polar Form of Complex Numbers. Powers and Roots . . . . . 97 13.3 Derivative. Holomorphic Function. . . . . . . . . . . . . . . . 100 13.4 Cauchy-Riemann Equations. Laplace’s Equation. . . . . . . . 103 13.5 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . 108 13.6 Trigonometric and Hyperbolic Functions . . . . . . . . . . . . 109 13.7 Logarithm. General Power . . . . . . . . . . . . . . . . . . . . 110 14 Complex Integration 113 14.1 Line Integral in the Complex Plane . . . . . . . . . . . . . . . 113 14.2 Cauchy’s Integral Theorem . . . . . . . . . . . . . . . . . . . . 121 14.3 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . 12514....
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.

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FullNotes - MATH 529 – Mathematical Methods for Physical Sciences II Christoph Kirsch Contents 0 Overview of the lecture 3 0.1 Partial

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