# Modulus of Exponential is Exponential of Real Part

## Theorem

Let $z \in \C$ be a complex number.

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

Let $\cmod {\, \cdot \,}$ denote the complex modulus

Then:

$\cmod {\exp z} = \map \exp {\Re z}$

where $\Re z$ denotes the real part of $z$.

## Proof

Let $z = x + iy$.

 $\displaystyle \cmod {\exp z}$ $=$ $\displaystyle \cmod {\map \exp {x + iy} }$ $\displaystyle$ $=$ $\displaystyle \cmod {\paren {\exp x} \paren {\exp i y} }$ Exponential of Sum $\displaystyle$ $=$ $\displaystyle \cmod {\exp x} \cmod {\exp i y}$ Modulus of Product $\displaystyle$ $=$ $\displaystyle \cmod {\exp x}$ Modulus of Exponential of Imaginary Number is One $\displaystyle$ $=$ $\displaystyle e^x$ Exponential of Real Number is Strictly Positive $\displaystyle$ $=$ $\displaystyle \map \exp {\Re z}$ Definition of Real Part

$\blacksquare$