Hyperbolic Cosine of Complex Number

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Theorem

Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.

Then:

$\cosh \paren {a + b i} = \cosh a \cos b + i \sinh a \sin b$

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

\(\displaystyle \map \cosh {a + b i}\) \(=\) \(\displaystyle \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}\) Hyperbolic Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \cosh a \cos b + \sinh a \map \sinh {b i}\) Cosine in terms of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \cosh a \cos b + i \sinh a \sin b\) Sine in terms of Hyperbolic Sine

$\blacksquare$


Proof 2

\(\displaystyle \cosh a \cos b - i \sinh a \sin b\) \(=\) \(\displaystyle \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2\) Definition of Hyperbolic Cosine, Cosine Exponential Formulation, Definition of Hyperbolic Sine, Sine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a + i b} + e^{-a + i b} + e^{a - i b} + e^{-a - i b} + e^{a + i b} - e^{-a + i b} - e^{a - i b} + e^{-a - i b} } 4\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^{a + i b} + e^{-\paren {a + i b} } } 2\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \cosh \paren {a + b i}\) Definition of Hyperbolic Cosine

$\blacksquare$


Also see