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Fourier_Series_Handout_2x2

Fourier_Series_Handout_2x2 - Fourier series Generalizations...

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Fourier series Generalizations Applications Chapter 11: Fourier Series Sections 1 - 5 Chapter 11: Fourier Series Fourier series Generalizations Applications Definition Convergence 1. Fourier series We saw before that the set of functions { 1 , cos( x ) , sin( x ), cos(2 x ) , sin(2 x ) , · · · , cos( mx ) , sin( mx ) , · · · } , where m is a non-negative integer, forms a complete orthogonal basis of the space of square integrable functions on [ - π , π ]. This means that we can define the Fourier series of any square integrable function on [ - π , π ] as f ( x ) = a 0 + n =1 [ a n cos( nx ) + b n sin( nx )] , where a 0 = 1 2 π π - π f ( x ) dx and, for n 1, a n = 1 π π - π f ( x ) cos( nx ) dx and b n = 1 π π - π f ( x ) sin( nx ) dx . Chapter 11: Fourier Series Fourier series Generalizations Applications Definition Convergence Convergence of Fourier series If f is continuously di ff erentiable on [ - π , π ] except at possibly a finite number of points where it has a left-hand and a right-hand derivative, then the partial sum f N ( x ) = a 0 + N n =1 [ a n cos( nx ) + b n sin( nx )] with the a i defined above, converges to f ( x ) as N → ∞ if f is continuous at x . At a point of discontinuity , the Fourier series converges towards 1 2 f ( x + ) + f ( x - ) .
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