# What is the relationship between different definitions of Fourier transform?

I always see various definitions of Fourier transform. A standard form is: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ and its attached inversion is $$f(x)=\int_{\mathbb{R}^d}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi$$ Another form is like this: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-ix\cdot\xi}dx$$ and the inversion formula is $$f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}\hat{f}(\xi)e^{ix\cdot\xi}d\xi$$

I believe they are actually the same and I try to find their relationship.

There exist several slightly different definitions of the Fourier transform which are commonly used; they differ in the choice of the constant 2π inside the exponential and/or a multiplicative constant before the integral. Their properties are essentially the same, and by a simple change of variable one can always translate statements using one of the definitions into statements using another one.

So I tried change of variable: $$\int_{\mathbb{R}^d}f(x)e^{-ix\cdot\xi}dx=(2\pi)^d\int_{\mathbb{R}^d}f(2\pi x)e^{-2\pi ix\cdot\xi}dx$$ But then since the variable in $f$ is $2\pi ix$ rather than $x$, I don't know how to deal with it. Can you please help? Thank you.

EDIT: According to James Edward Lewis, the change of variable should be $2\pi x$. I revised this.

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The difference between the first and the second form is in the definition of $\xi$ (so you should change the name of the variables correspondingly). Let me introduce the new notation $\tilde\xi$ and $\tilde f(\tilde \xi)$ such that the confusion is (hopefully) lifted.

You want to proof that $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx \qquad f(x)=\int_{\mathbb{R}^d}\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi$$ is equivalent to $$\tilde{f}(\tilde\xi)=\int_{\mathbb{R}^d}f(x)e^{-ix\cdot\tilde\xi}dx \qquad f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}\tilde{f}(\tilde\xi)e^{ix\cdot\tilde\xi}d\tilde\xi.$$

Let's take the first form as given and try to derive the second form. The Fourier transform is by definition the same with $\tilde\xi = 2\pi \xi$ and $\tilde f(\tilde \xi) = \hat{f} (\xi)$. For the inversion formula, we have to work a bit harder (and use substitution). $$f(x)=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}\tilde{f}(\tilde\xi)e^{ix\cdot\tilde\xi}d\tilde\xi = \frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}\tilde{f}(\tilde\xi)e^{2\pi ix\cdot\xi}d(2 \pi \xi) = \int_{\mathbb{R}^d}\hat{f}(\xi)e^{2\pi ix\cdot\tilde\xi}d\xi.$$

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