The Fourier Transform t*g(t)

Fourier Transform Theory
Fourier Transform Pairs
Fourier Transform Applications

This page will show us how to get the Fourier Transform of an arbitrary function g(t) multiplied by t. We just need to ensure that G(f) exists, and this result holds. This can be done with a simple trick involving interchanging the order of integration and differentiation. Start with a function g(t) and it's Fourier Transform G(f) and take the derivative with respect to frequency:

derivation of fourier transform for t*g(t)
[Equation 1]

Hence, we can re-arrange terms in Equation [1] to get the final result:

result of fourier transform for t*g(t)
[Equation 2]

It's that simple! This can be used to derive other Fourier Transforms.

We can use the same trick to find the Fourier Transform for t^n * h(t). Just assume n is a positive integer, and that the Fourier Transform of h(t) is H(f). Then:

derivation or proof of fourier transform
[Equation 3]

And the result follows:

derivation or proof of fourier transform for t^n*f(t)
[Equation 4]

Fourier Transform Pairs Table of Contents

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