Fourier transform as Laplace transform on the imaginary axis

Summary

The Fourier transform of a function on $\R$ is the two-sided Laplace transform 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.