# Imaginary 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:

$\map \Im {\exp z} = e^x \sin y$

where:

$\map \Im z$ denotes the imaginary part of a complex number $z$
$e^x$ denotes the real exponential function of $x$
$\sin y$ denotes the real sine 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 imaginary part of a complex number.

$\blacksquare$