Even Impulse Pair is Fourier Transform of Cosine Function

Theorem
Consider the (real) cosine function $\map \cos x: \R \to \R$.


 * $\map f x = \map {\cos} {\pi x }$

Then:

where:
 * $\map {\hat f} s$ is the Fourier transform of $\map f x$.
 * $\operatorname {II}$ denotes the even impulse pair function.

Proof
By the definition of a Fourier transform: