# 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, ${\displaystyle f(t)}$  for example, to the function which has variable of frequency.

## Definition

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(t)\ e^{-i2\pi t\xi }\,dt}$ ...(1)

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

On the other hand, Inverse Fourier transform is defined as follows:

${\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{i2\pi t\xi }\,d\xi }$  ...(2)

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

1.

${\displaystyle {\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt}$
${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(t)e^{i\omega t}d\omega }$

2.

${\displaystyle {\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt}$
${\displaystyle f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {f}}(t)e^{i\omega t}d\omega }$