Fourier Transform of Exponential Function

Theorem
Let $\map f x$ be defined as the Real Exponential function where the Absolute Value of the input is used in the exponent and the exponent is scaled by a factor of $-2 \pi a$:


 * $\map f x = e^{-2 \pi a \size x}:   \R \to \R$

Then:


 * $\map {\hat f} s = \dfrac 1 \pi \dfrac a {a^2 + s^2}$

where $\map {\hat f} s$ is the Fourier transform of $\map f x$.

Proof
By the definition of a Fourier transform:

Therefore:


 * $\map {\hat f} s = \dfrac 1 \pi \dfrac a {a^2 + s^2 }$