Hyperbolic Cosine of Complex Number/Proof 1
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Theorem
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
- $\map \cosh {a + b i} = \cosh a \cos b + i \sinh a \sin b$
where:
- $\cos$ denotes the real cosine function
- $\sin$ denotes the real sine function
- $\sinh$ denotes the hyperbolic sine function
- $\cosh$ denotes the hyperbolic cosine function
Proof
\(\ds \map \cosh {a + b i}\) | \(=\) | \(\ds \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}\) | Hyperbolic Cosine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh a \cos b + \sinh a \map \sinh {b i}\) | Cosine in terms of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh a \cos b + i \sinh a \sin b\) | Sine in terms of Hyperbolic Sine |
$\blacksquare$