# c21 - 18.03 Class 21 April 3 Fun with Fourier series[1 If...

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Fun with Fourier series [1] If f(t) is any decent periodic of period 2pi, it has exactly one expression as f(t) = (a0/2) + a1 cos(t) + a2 cos(2t) + . .. (*) + b1 sin(t) + b2 sin(2t) + . .. To be precise, there is a single list of coefficients such that this is true for every value of t = a for which f(t) is continuous at a. The coefficients can be computed by the integral formulas a_n = (1/pi) integral_{-pi}^pi f(t) cos(nt) dt b_n = (1/pi) integral_{-pi}^pi f(t) sin(nt) dt but one can often discover them without evaluating these integrals. [2] Example: the "standard squarewave" sq(t) = 1 for 0 < t < pi, -1 for -pi < 0 < 0 has Fourier series sq(t) = (4/pi) sum_{n odd} (sin(nt))/n as we saw by calculating the integrals. Let's review that: Any odd function has a sine series -- a_n = 0 for all n -- and the b_n's can be computed using the simpler integral b_n = (2/pi) int_0^pi f(t) sin(nt) dt In that range sq(t) = 1 , so we must compute int_0^pi sin(nt) dt = - (1/n) cos(nt) |_0^pi = - (1/n) [ cos(n pi) - 1 ] Now the graph of cos(t) shows that n | cos(n pi) | 1 - cos(n pi) ------------------------------------------ 0 | 1 | 0 1 | -1 | 2 2 | 1 | 0 ... ... ... so b_n = (4/n pi) if n is odd, = 0 if n is even. sq(t)

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## This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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c21 - 18.03 Class 21 April 3 Fun with Fourier series[1 If...

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