This page contains various complex hyperbolic functions, including Sinh, Cosh, Tanh functions.
Hyperbolic Sine Function
It is known that a complex Sinh function can be calculated using complex exponential functions:
sinh z = 1/2 (e^{z}  e^{z})
= 1/2 [e^{x} (cos y + i siny)  e^{x} (cos y  i sin y)]
= sinh x cos y + i cosh x sin y

Using this relationship, we wrote the following method:
public static Complex Sinh(Complex c)
{
return new Complex(Math.Sinh(c.Real) * Math.Cos(c.Imaginary) +
(Math.Cosh(c.Real) * Math.Sin(c.Imaginary));
}
Hyperbolic Cosine Function
Similar to the case of the complex Sinh function, a complex Cosh function can also be represented by:
cosh z = 1/2 (e^{z} + e^{z})
= 1/2 [e^{x} (cos y + i siny) + e^{x} (cos y  i sin y)]
= cosh x cos y + i sinh x sin y

Using the above equation we added this method to the "Complex class". (Note  Complex class is part of our
mathematical algorithms library and is not shown in these examples):
public static Complex Cosh(Complex c)
{
return new Complex(Math.Cosh(c.Real) * Math.Cos(c.Imaginary) +
(Math.Sinh(c.Real) * Math.Sin(c.Imaginary));
}
Hyperbolic Tangent Function
A complex tanh can be computed using Sinh and Cosh functions:
Adding a complex Tanh method to the Complex class:
public static Complex Tanh(Complex c)
{
return Sinh(c) / Cosh(c);
}
