# Definition:Fourier Transform/Real Function/Formulation 3

## Definition

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

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

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

## 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.

## Sources

- 1978: Ronald N. Bracewell:
*The Fourier Transform and its Applications*(2nd ed.) ... (previous) ... (next): Chapter $2$: Groundwork: The Fourier transform and Fourier's integral theorem - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Fourier transform** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Fourier transform**