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285lect32 - 1 Lecture 33 1.1 Fourier coe¢ cients more...

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Unformatted text preview: 1 Lecture 33 1.1 Fourier coe¢ cients: more formulae Notice that because f ( t ) ; sin nt and cos nt are periodic functions, a m = 1 & R & + ¡ & & + ¡ f ( t ) cos mtdt; m = 0 ; 1 ; 2 ; ::: , b m = 1 & R & + ¡ & & + ¡ f ( t ) sin mtdt; m = 1 ; 2 ; ::: . In particular, for & = ¡ , we get a m = 1 & R 2 & f ( t ) cos mtdt; m = 0 ; 1 ; 2 ; ::: , b m = 1 & R 2 & f ( t ) sin mtdt; m = 1 ; 2 ; ::: . Recall that a function as called even if, for every t , f ( t ) = f ( & t ) : For even function, Z a & a f ( t ) dt = 2 Z a f ( t ) dt: f ( t ) is called odd if f ( t ) = & f ( & t ) : For odd function, Z a & a f ( t ) dt = 0 : If f ( t ) is an even function, then f ( t ) cos nt also is an even function and f ( t ) sin nt is an odd function. Therefore, for even function we have the following formulas for Fourier coe¢ cients: a n = 2 ¡ Z & f ( t ) cos ntdt , n = 0 ; 1 ; 2 ; ::: and b n = 0 ; n = 1 ; 2 ; ::: ....
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