# Theorem On Inverse Fourier Transform

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**Theorem on Inverse Fourier Transform:**

Theorem. Let f ∈ L^{1} (R) and let f be piecewise smooth on R. Then for every x ∈R,

If x is a point of continuity of f, then

For all Φ ∈ L^{1} (R), we call The inverse Fourier transform of Φ = f^, then we define;

at a point of discontinuity of f.

Then That is F^{-1}Ff = f.

**Examples:
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The inverse Fourier transform is useful in computing Fourier transforms. Since It follows that

Interchanging the role of x and ξ multiplying by 2π leads to In a similar functions every Fourier Series transform pair defines a dual pair using the Inverse Fourier Transform.