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Unformatted text preview: Even and odd functions • We define an even function such that f ( x ) = f ( x ) • We define an odd function such that f ( x ) = f ( x ) • Example, sin x is an odd function because sin x = sin x • Example, cos x is an even function because cos x = cos x • Now consider a Fourier series of a periodic, even function f ( x ) ( f ( x ) = f ( x )), over the interval π < x < π f ( x ) = a 2 + ∞ X n =1 a n cos nx + ∞ X n =1 b n sin nx • Now consider the integrals to determine the coefficients, first a a = 1 π Z π π f ( x ) dx = 2 π Z π f ( x ) dx Even/odd functions continued • Next the a n for finite n , we again use the fact that f ( x ) is even, and also cos nx is even, a n = 1 π Z π π f ( x ) cos nxdx = 2 π Z π f ( x ) cos nxdx • Next we can show b n = 0 when f ( x ) is even, b n = 1 π Z π f ( x ) sin nxdx Z π f ( x ) sin nxdx = 0 Even/odd functions continued • We can also treat the odd case f ( x ) = f ( x ), then a = 1 π Z π π f ( x ) dx = 0 a n = 1 π Z π f ( x ) cos nxdx Z π f ( x ) cos nxdx = 0 b n = 1 π Z π π f ( x ) sin nxdx = 2 π Z π f ( x ) sin nxdx Example of an odd function • The step function provides us an example of an odd function...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
 Staff

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