# 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:

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

### Formulation 2

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

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

### Formulation 3

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

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

## Also denoted as

The **Fourier transform of $f$** is also often seen denoted by $\hat f$, as this notation has considerable usefulness.

The style of $\FF$ can vary across different sources. $\mathsf{Pr} \infty \mathsf{fWiki}$ uses $\FF$ as standard.

Some sources write $\FF \sqbrk f$ instead of $\map \FF f$.

Some sources omit the brackets altogether, and deploy it as $\FF f$.

## 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.