Applied Mathematics/General Fourier Transform

Fourier Transform
Fourier Transform is to transform the function which has certain kinds of variables, such as time or spatial coordinate, $$f(t)$$ for example, to the function which has variable of frequency.

Definition

 * $$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\ e^{- i 2\pi t \xi}\,dt$$...(1)

This integral above is referred to as Fourier integral, while $$\hat{f}(\xi)$$ is called Fourier transform of $$f(t)$$. $$t$$ denotes "time". $$\xi$$ denotes "frequency".

On the other hand, Inverse Fourier transform is defined as follows:
 * $$f(t) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{i 2\pi t \xi}\,d\xi$$ ...(2)

In the textbooks of universities, the Fourier transform is usually introduced with the variable Angular frequency $$\omega$$. In other word, $$\xi \rightarrow \omega = 2 \pi \xi$$ is substituted to (1) and (2) in the books. In that case, the Fourier transform is written in two different ways.

1.
 * $$\hat{f}(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$$
 * $$f(t)= \frac{1}{2\pi} \int_{-\infty}^{\infty}\hat{f}(t)e^{i\omega t}d\omega$$

2.
 * $$\hat{f}(\omega)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$$
 * $$f(t)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}\hat{f}(t)e^{i\omega t}d\omega$$