Hyperbolic Cosine in terms of Cosine

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Theorem

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

Then:

$\cosh z = \map \cos {i z}$

where:

$\cos$ denotes the complex cosine
$\cosh$ denotes the hyperbolic cosine
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds \map \cos {i z}\) \(=\) \(\ds \frac {e^{i \paren {i z} } + e^{-i \paren {i z} } } 2\) Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac {e^{-z} + e^z} 2\) $i^2 = -1$
\(\ds \) \(=\) \(\ds \cosh z\) Definition of Hyperbolic Cosine

$\blacksquare$


Also see


Sources

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