Imaginary Part of Complex Exponential Function
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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.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$