11.Fourier Transforms.ppt

Chapter 11 Fourier Series, Integrals, and Transforms 11.1 Fourier Series 11.2 Functions of Any Period p = 2L 11.3 Even and Odd Functions. Half-Range Expansions 11.4 Complex Fourier Series. Optional 11.5 Forced Oscillations 11.6 Approximation by Trigonometric Polynomials 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 Trigonometric Series Periodic Function： f(x+p)=f(x) p：period of f(x) Orthogonality of the Trigonometric System The trigonometric system: Euler Formulas for the Fourier Coefficients Convergence and Sum of Fourier Series Theorem1：Representation by a Fourier series §11.5 Forced Oscillations Example § 11.8 Fourier Cosine and Sine Transforms Fourier Cosine Transforms Fourier Sine Transforms * * §10.1 Fourier Series Trigonometric function is also Periodic Function if the series converges, its sum will be a function of period 2π Where Explain = twice the average of f(x) cos nx in the period = twice the average of f(x) sin nx in the period = the average of f(x) in the period DIRICHLET’S THEOREM §10.2 Functions of Any Period p=2L Euler formulas = the average in the period = twice the average in the period = twice the average in the period § 10.3 Even and Odd Functions Half-Range Expansions Even and Odd Functions： The Fourier series of an even function of period 2L is a “Fourier cosine series” Theorem 1a：Fourier cosine series = the average in the half period = twice the average in the half period The Fourier series of an odd function of period 2L is a “Fourier sine series” Theorem 1b：Fourier sine series = twice the average in the half period Theorem 2： Sum of functions §11.4 Complex Fourier Series in terms of angle whose units are radians in terms of length (11.4.1-2) (11.4.1-1) (11.4.2-1) (11.4.2-2) Rewrite as Where is Ke (11.4.4-1) (11.4.4-2) (11.4.3) normalized orthogonal functions i.e. orthonor