Hyperbolic Cosine of Complex Number
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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 1
\(\ds \map \cosh {a + b i}\) | \(=\) | \(\ds \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}\) | 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 |
$\blacksquare$
Proof 2
\(\ds \cosh a \cos b - i \sinh a \sin b\) | \(=\) | \(\ds \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2\) | Definition of Hyperbolic Cosine, Euler's Cosine Identity, Definition of Hyperbolic Sine, Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a + i b} + e^{-a + i b} + e^{a - i b} + e^{-a - i b} + e^{a + i b} - e^{-a + i b} - e^{a - i b} + e^{-a - i b} } 4\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a + i b} + e^{-\paren {a + i b} } } 2\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \cosh \paren {a + b i}\) | Definition of Hyperbolic Cosine |
$\blacksquare$