---
tag: J7Z1H328BC
kind: concept
name: Fourier transform as Laplace transform on the imaginary axis
msc: [44A10, 42A38]
sources:
  - url: https://en.wikipedia.org/wiki/Two-sided_Laplace_transform
---

## Summary

The [[Fourier transform|DPVM34ADZV]] of a function on $\R$ is the [[two-sided Laplace transform|6J2AGD0BYY]] of the same function evaluated on the imaginary axis.
With $\hat f(\xi) = \int f(x) e^{-2\pi i x \xi} \, dx$, this reads $\hat f(\xi) = \mathcal{B}f(2\pi i \xi)$.
More generally, the values of $\mathcal{B}f$ on a vertical line $\operatorname{Re} s = \sigma$ are the Fourier transform of $e^{-\sigma x} f(x)$.
This lets results on one transform pass to the other, such as inversion formulas, Plancherel's theorem and the use of analytic continuation.

## Explanations

### L1 human

The Laplace transform uses the weights $e^{-sx}$ with a complex number $s$.
When $s$ is purely imaginary, these weights are the pure oscillations $e^{-i\omega x}$ of the Fourier transform.
So the Fourier transform is the Laplace transform seen along one vertical line of the complex plane.
For example, $f(x) = e^{-\abs{x}}$ has $\mathcal{B}f(s) = 2/(1 - s^2)$ for $\abs{\operatorname{Re} s} < 1$, and putting $s = 2\pi i \xi$ gives $\hat f(\xi) = 2/(1 + 4\pi^2 \xi^2)$.