Modulus of Exponential is Exponential of Real Part

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

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

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


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

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


Let $z = x + iy$.

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