Just began a serious study of the Fourier transform with a couple of books. One of them defines the Fourier transform on [itex]\mathbb R[/itex] as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\hat f(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\xi x}dx.

[/tex]

Another defines it as

[tex]

\hat f(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i \xi x} dx.

[/tex]

A few questions:

(1) Are these definitions somehow equivalent? I cannot seem to obtain the second from the first by making a simple change of variables.

(2) Why worry about the factors of [itex]2\pi[/itex] in the definitions? What does that do for us? Why not leave those out altogether?

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# Competing definitions of the Fourier transform

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