Hyperbolic Tangent of Complex Number/Formulation 4

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Theorem

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

Let $i$ be the imaginary unit.


Then:

$\map \tanh {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$

where:

$\tanh$ denotes the hyperbolic tangent function
$\sin$ denotes the real sine function
$\cos$ denotes the real cosine function
$\sinh$ denotes the hyperbolic sine function
$\cosh$ denotes the hyperbolic cosine function.


Proof

\(\ds \map \tanh {a + b i}\) \(=\) \(\ds \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}\) Hyperbolic Tangent of Complex Number: Formulation 1
\(\ds \) \(=\) \(\ds \dfrac {\paren {\sinh a \cos b + i \cosh a \sin b} \paren {\cosh a \cos b - i \sinh a \sin b} } {\paren {\cosh a \cos b + i \sinh a \sin b} \paren {\cosh a \cos b - i \sinh a \sin b} }\)
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cosh a \cos^2 b + i \cosh^2 a \cos b \sin b - i \sinh^2 a \cos b \sin b + \sinh a \cosh a \sin^2 b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cosh a \paren {\cos^2 b + \sin^2 b} + i \paren {\cosh^2 a - \sinh^2 a} \cos b \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cosh a + i \paren {\cosh^2 a - \sinh^2 a} \cos b \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sinh a \cosh a + i \cos b \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \dfrac {\sinh 2 a + i \sin 2 b} {2 \paren {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b} }\) Double Angle Formula for Hyperbolic Sine, Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds \dfrac {\sinh 2 a + i \sin 2 b} {\cosh^2 a \paren {1 + \cos 2 b} + \sinh^2 a \paren {1 - \cos 2 b} }\) Double Angle Formula for Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sinh 2 a + i \sin 2 b} {\cosh^2 a + \cosh^2 a \cos 2 b + \sinh^2 a - \sinh^2 a \cos 2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \paren {\cosh^2 a - \sinh^2 a} \cos 2 b}\) Double Angle Formula for Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}\) Difference of Squares of Hyperbolic Cosine and Sine

$\blacksquare$


Also see


Sources

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