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.