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:
- $\cmod {\exp z} = \map \exp {\map \Re z}$
where $\map \Re z$ denotes the real part of $z$.
Proof
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 |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$