# Definition:Fourier Transform/Real Function

## Definition

Let $f: \R \to \R$ be a real function which is Lebesgue integrable.

### Formulation 1

The **Fourier transform** of $f$ is defined and denoted as:

- $\displaystyle \map \FF {\map f t} = \map F s := \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$

### Formulation 2

The **Fourier transform** of $f$ is defined and denoted as:

- $\displaystyle \map \FF {\map f t} = \map F s := \int_{-\infty}^\infty e^{-i s t} \map f t \rd t$

### Formulation 3

The **Fourier transform** of $f$ is defined and denoted as:

- $\displaystyle \map \FF {\map f t} = \map F s := \dfrac 1 {\sqrt {2 \pi} } \int_{-\infty}^\infty e^{-i s t} \map f t \rd t$

## Also known as

The real Fourier transform function is sometimes referred to as the **minus-$i$ transform of $\map f t$**.

This allows us to distinguish between this and the real inverse Fourier transform function, known in turn as the **plus-$i$ transform of $\map F s$**.

Ronald N. Bracewell, in his *The Fourier Transform and its Applications, 2nd ed.* of $1978$, discusses all $3$ of the formulations given in $\mathsf{Pr} \infty \mathsf{fWiki}$, referring to them as **System $1$**, **System $2$** and **System $3$**.

The numbers assigned to Formulation $1$, Formulation $2$ and Formulation $3$ have been configured so as to correspond to these directly.

## Also see

- Results about
**Fourier transforms**can be found here.

## Source of Name

This entry was named for Joseph Fourier.