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I have been learning about Fourier Transform and its derivations. My question is regarding to the final part of the derivation in the textbook I'm reading. It goes like:

$f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\left [\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }f(u)e^{-iwu} du\right ]e^{iwx}dw$

We define the expression inside the square bracket as Fourier Transform $\hat{f}$ and therefore $\hat{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }f(x)e^{-iwx} dx$

I understand that $w$ is the frequency specturm which is the input, but it felt very arbitrary that we chose to define the Fourier transform function in such a way. Why exactly did we select the section inside square bracket? What does the output represent?

Thank you.

Disintegrating By Parts
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An537
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    Welcome to MSE. Please don't post text-only pictures. – Another User Aug 03 '22 at 16:50
  • What is "that specific part of the complex Fourier integral?" – Sean Roberson Aug 03 '22 at 17:01
  • Sorry I'm not very familiar with the platform. I have edited the question for clarity. – An537 Aug 04 '22 at 07:49
  • The definitions are rather arbitrary. The Fourier and inverse Fourier transforms can be defined more generally (see formulas (15) and (16) at https://mathworld.wolfram.com/FourierTransform.html) and their definitions can be interchanged by changing the sign of the Fourier parameter $b$ when the Fourier parameter $a=0$ which is the case here. – Steven Clark Aug 04 '22 at 16:53
  • I don't know why people here are so slap-happy about -1 increments added to a question. Asking about the normalization is a good question. Where does the normalization come from? It basically had to come out of a derivation of some kind; so what is the derivation that gives you the proper normalization? Maybe someone can point you to a reference. – Disintegrating By Parts Aug 15 '22 at 18:53
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    @DisintegratingByParts One downvote is not unusual at all and no reason to blame the community. If there were several ones, I would understand your angriness. I received many downvotes (in most questions a single one) beyond ridicoulous. I do not understand this one either and upvoted. – Peter Sep 09 '22 at 08:56
  • @Peter : The new user has 21 points, and I tend to think that slapping a downvote on a question from a new user is not productive, unless you're going to leave some kind of clarification. Thanks for up-voting as well. – Disintegrating By Parts Sep 09 '22 at 09:17
  • The question of normalization is a good one. One way to get at it is through Complex Analysis through the $2\pi i$ factor in the Cauchy integral. There are derivations here (including one of mine using complex analysis): https://math.stackexchange.com/questions/2872415/a-list-of-proofs-of-fourier-inversion-formula – Disintegrating By Parts Sep 10 '22 at 05:50

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