Real Part of Complex Exponential Function

Theorem
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.

Let $\exp z$ denote the complex exponential function.

Then:
 * $\Re \paren {\exp z} = e^x \cos y$

where:
 * $\Re z$ denotes the real part of a complex number $z$
 * $e^x$ denotes the real exponential function of $x$
 * $\cos y$ denotes the real cosine function of $y$.

Proof
From the definition of the complex exponential function:
 * $\exp z := e^x \paren {\cos y + i \sin y}$

The result follows by definition of the real part of a complex number.