Hyperbolic Tangent of Complex Number/Formulation 1

Theorem

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

Let $i$ be the imaginary unit.

Then:

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

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

\(\ds \tanh \paren {a + b i}\)

\(=\)

\(\ds \frac {\sinh \paren {a + b i} } {\cosh \paren {a + b i} }\)

Definition of Hyperbolic Tangent

\(\ds \)

\(=\)

\(\ds \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}\)

Hyperbolic Sine of Complex Number and Hyperbolic Cosine of Complex Number

$\blacksquare$

Also see

• Hyperbolic Sine of Complex Number
• Hyperbolic Cosine of Complex Number
• Hyperbolic Cosecant of Complex Number
• Hyperbolic Secant of Complex Number
• Hyperbolic Cotangent of Complex Number