Modulus of Exponential is Exponential of Real Part

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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

$\blacksquare$


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