Thus we nd the fourier series for f f t 4 sin t 4 4

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Unformatted text preview: sics of Fourier analysis in its simplest context. A function f : R → R is said to be periodic of period T if it satisfies the relation f (t + T ) = f (t), for all t ∈ R . Thus f (t) = sin t is periodic of period 2π . Given an arbitrary period T , it is easy to construct examples of functions πt which are periodic of period T —indeed, the function f (t) = sin( 2T ) is periodic of period T because sin( 2π (t + T ) 2πt 2πt ) = sin( + 2π ) = sin( ). T T T More generally, if k is any positive integer, the functions cos( 2πkt ) T and sin( 2πkt ) T are also periodic functions of period T . The main theorem from the theory of Fourier series states that any “wellbehaved” periodic function of period T can be expressed as a superposition of sines and cosines: f (t) = a0 2πt 4πt 2πt 4πt + a1 cos( ) + a2 cos( ) + . . . + b1 sin( ) + b2 sin( ) + ... . 2 T T T T (3.1) In this formula, the ak ’s and bk ’s are called the Fourier coefficients of f , and the infinite series on the right-hand side is called the Fourier series of f . Our first goal is to determine how to calculate thes...
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This document was uploaded on 01/12/2014.

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