Change of variable of Fourier Transform in Tim Olson's Book

Asked Apr 17 '21 at 13:03

Active Apr 17 '21 at 13:03

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I saw in Tim Olson's Applied Fourier Analysis book this suposed variable change, where he says there is two equivalent ways of writing the fourier transform of a function:

$$ \hat{f}(s) = \frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} f(t) e^{ist}dt \tag{1}\label{l1} $$

$$ \hat{f}(s) = \int^\infty_{-\infty} f(t) e^{-2\pi ist}dt \tag{2}\label{l2} $$

So he basically says from $\ref{l1} $ to $\ref{l2}$ there is just a simple variable change.

I couldn't see how he achieved that. Maybe something obvious I'm not seeing.

I've tried integration by parts but didn't seem to work well...

Any hints? Is his claim correct?

Thanks in advance

asked Apr 17 '21 at 13:03

Pablo Carneiro Elias

1

Have you looked at the Wikipedia page? Also see this answer. – Axion004 Apr 17 '21 at 15:32
Thanks. It seems there is still a minus sign that is not correct though. – Pablo Carneiro Elias Apr 17 '21 at 23:18
1

Did you read the "Summary of popular forms of the Fourier transform, one-dimensional" part of the Wikipedia page? It describes the appropriate change of variable (as does the second most upvoted answer in the link above). If necessary, I could write out an answer to your question (even though it is now a couple of months old). – Axion004 Aug 05 '21 at 21:32

Change of variable of Fourier Transform in Tim Olson's Book

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