fourierFormulas

fourierFormulas - -∞ ˆ f w e iwx dw f x = F-1 c ˆ f c =...

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Fourier series/transform information for Exam 2. Fourier series: Let f ( x ) be a 2 L -periodic function. Then f ( x ) = a 0 + X n =1 ± a n cos ± nπx L ² + b n sin ± nπx L ²² where a 0 = 1 2 L Z L - L f ( x ) dx a n = 1 L Z L - L f ( x ) cos ± nπx L ² dx b n = 1 L Z L - L f ( x ) sin ± nπx L ² dx Fourier transforms: With appropriate conditions on f , ˆ f ( w ) = F ( f ( x )) = 1 2 π Z -∞ f ( x ) e - iwx dx ˆ f c ( w ) = F c ( f ( x )) = r 2 π Z 0 f ( x ) cos wxdx ˆ f s ( w ) = F s ( f ( x )) = r 2 π Z 0 f ( x ) sin wxdx These satisfy f ( x ) = F - 1 ( ˆ f ) = 1 2 π Z
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Unformatted text preview: -∞ ˆ f ( w ) e iwx dw f ( x ) = F-1 c ( ˆ f c ) = r 2 π Z ∞ ˆ f c ( w ) cos wxdw f ( x ) = F-1 s ( ˆ f s ) = r 2 π Z ∞ ˆ f s ( w ) sin wxdw F ( f ) = iw ˆ f ( w ) F c ( f ) = w ˆ f s ( w )-r 2 π f (0) F s ( f ) =-w ˆ f c ( w )...
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