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11 Fourier Analysis
11.1 Fourier Series
11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 
11.3 Forced Oscillations 
11.4 Approximation by Trigonometric Polynomials 
11.5 Sturm–Liouville Problems. Orthogonal Functions 
11.6 Orthogonal Series. Generalized Fourier Series 
11.7 Fourier Integral 
11.8 Fourier Cosine and Sine Transforms 
11.9 Fourier Transform. Discrete and Fast Fourier Transforms 
11.10 Tables of Transforms 

12 Partial Differential Equations (PDEs) 
12.1 Basic Concepts of PDEs
12.2 Modeling: Vibrating String, Wave Equation 
12.3 Solution by Separating Variables. Use of Fourier Series 
12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 
12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 
12.6 Heat Equation: Solution by Fourier Series Steady Two-Dimensional Heat Problems. Dirichlet Problem 
12.7 Heat Equation: Modeling Very Long Bars Solution by Fourier Integrals and Transforms 
12.8 Modeling: Membrane, Two-Dimensional Wave Equation 
12.9 Rectangular Membrane. Double Fourier Series 
12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 
12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 
12.12 Solution of PDEs by Laplace Transforms

13 Complex Numbers and Functions Complex Differentiation 
13.1 Complex Numbers and Their Geometric Representation 
13.2 Polar Form of Complex Numbers. Powers and Roots 
13.3 Derivative. Analytic Function 
13.4 Cauchy–Riemann Equations. Laplace’s Equation 
13.5 Exponential Function 
13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 
13.7 Logarithm. General Power. Principal Value 

14 Complex Integration 
14.1 Line Integral in the Complex Plane 
14.2 Cauchy’s Integral Theorem 
14.3 Cauchy’s Integral Formula 
14.4 Derivatives of Analytic Functions 

15 Power Series, Taylor Series 
15.1 Sequences, Series, Convergence Tests 
15.2 Power Series 
15.3 Functions Given by Power Series 
15.4 Taylor and Maclaurin Series 
15.5 Uniform Convergence. Optional 

16 Laurent Series. Residue Integration 
16.1 Laurent Series
16.2 Singularities and Zeros. Infinity 
16.3 Residue Integration Method 
16.4 Residue Integration of Real Integrals 

17 Conformal Mapping 
17.1 Geometry of Analytic Functions: Conformal Mapping
17.2 Linear Fractional Transformations (Möbius Transformations) 
17.3 Special Linear Fractional Transformations 
17.4 Conformal Mapping by Other Functions 
17.5 Riemann Surfaces. Optional 

18 Complex Analysis and Potential Theory 
18.1 Electrostatic Fields 
18.2 Use of Conformal Mapping. Modeling 
18.3 Heat Problems 
18.4 Fluid Flow 
18.5 Poisson’s Integral Formula for Potentials 
18.6 General Properties of Harmonic Functions.