Hyperbolic Sine of Complex Number

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

Let $i$ be the imaginary unit.

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

where:
 * $\sin$ denotes the real sine function
 * $\cos$ denotes the real cosine function
 * $\sinh$ denotes the hyperbolic sine function
 * $\cosh$ denotes the hyperbolic cosine function.

Also see

 * Hyperbolic Cosine of Complex Number
 * Hyperbolic Tangent of Complex Number
 * Hyperbolic Cosecant of Complex Number
 * Hyperbolic Secant of Complex Number
 * Hyperbolic Cotangent of Complex Number