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Hyperbolic Functions

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 (ez - e-z)
= 1/2 [ex (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 (ez + e-z)
= 1/2 [ex (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:

tanh z = sinh z / cosh z

Adding a complex Tanh method to the Complex class:

           public static Complex Tanh(Complex c)
             return Sinh(c) / Cosh(c);

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