15.3 Fourier Integral. Fourier Integral In ch 12, f(x) defined in (-p,p) f(x)= FS(x) Fourier Series (periodic function) In 15.3, f(x) defined in (-infinity,

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Presentation transcript:

15.3 Fourier Integral

Fourier Integral In ch 12, f(x) defined in (-p,p) f(x)= FS(x) Fourier Series (periodic function) In 15.3, f(x) defined in (-infinity, + infinity) f(x)= FI(x) Fourier Integral

Fourier Integral Definition f(x) defined in (-infinity, + infinity) The Fourier Integral f(x) is given by where Example1: Find the Fourier Integral representation of the function

Remarks 1) f(x) even -  f(x)sin odd -  B( )=0 -  Fourier cosine Integral FCI where 2) f(x) odd -  f(x)cos odd -  A( )=0 -  Fourier sine Integral FSI where 3) FI (x) = f(x) where x is a point of continuity FI (x) = average where x is a point of discontinuity 4) f(x) defined on ( 0, inf ) then FCI or FSI Example1: represent f(x) = exp(-x), x > 0 a) by cosine integral b) by sine integral

Complex Fourier Integral Definition f(x) defined in (-infinity, + infinity) The Fourier Integral in complex form of f(x) is given by where