Definition:Inverse Fourier Transform/Real Function/Formulation 2
Definition
Let $F: \R \to \R$ be a real function which is Lebesgue integrable.
The inverse Fourier transform of $F$ is defined and denoted as:
- $\ds \map \FF {\map F s} := \map f t = \dfrac 1 {2 \pi} \int_{-\infty}^\infty e^{i t s} \map F s \rd s$
Also known as
The real inverse Fourier transform function is sometimes referred to as the plus-$i$ transform of $\map F s$.
This allows us to distinguish between this and the real Fourier transform function, known in turn as the minus-$i$ transform of $\map f t$.
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 inverse Fourier transforms can be found here.
Source of Name
This entry was named for Joseph Fourier.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 33$: Fourier Transforms: Fourier Transforms: $33.8$
- 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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Fourier transform: 1.