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Second Order Linear Partial Differential Equations Part II Fourier series; EulerFourier formulas; Fourier Convergence Theorem; Even and odd functions; Cosine and Sine Series Extensions; Particular solution of the heat conduction equation Fourier Series Suppose f is a periodic function with a period T = 2L . Then the Fourier series representation of f is a trigonometric series (that is, it is an infinite series consists of sine and cosine terms) of the form ∞ a0  nπx nπ x  f x ) = +∑an cos +bn sin  2 n =1  L L  Where the coefficients are given by the EulerFourier formulas : L 1 mπ x am = L ∫ f x) cos L dx , m = 0, 1, 2, 3, … −L L 1 nπ x bn = L ∫ f x) sin L dx , n = 1, 2, 3, … −L The coefficients a’s are called the Fourier cosine coefficients (including a0, the constant term, which is in reality the 0th cosine term), and b’s are called the Fourier sine coefficients . © 2008, 2012 Zachary S Tseng E2 1 Note 1: Thus, every periodic function can be decomposed into a sum of one or more cosine and/or sine terms of selected frequencies determined solely by that of the original function. Conversely, by superimposing cosines and/ or sines of a certain selected set of