Complex Hyperbolic Function

The hyperbolic cosine and hyperbolic sine functions are:

Sinh(z) =

Cosh(z) =

We can easily create the other complex hyperbolic trigonometric functions.

tanh(z) =

coth(z) =

sech(z) =

csch(z) =

The derivatives of the hyperbolic functions are:

Sinh(z) = Cosh(z)

Cosh(z) = Sinh(z)

tanh(z) = sech(z)²

coth(z) = - csch(z)²

sech(z) = - sech(z)tanh(z)

csch(z) = - csch(z)coth(z)

The hyperbolic cosine and hyperbolic sine can be expressed as:

csch(z) = csch(x+iy) =
= Cos(y)csch(x) + iSinh(x)Sin(y)

Sinh(z) = Sinh(x+iy) =
= Sinh(x)Cos(y) + iCosh(x)Sin(y)

Some of the important identities involving the hyperbolic functions are:

Cosh(z)² - Sinh(z)² = 1

Sinh(z₁+ z₂) = Sinh(z₁)Cosh(z₂) + Cosh(z₁)Sinh(z₂)

Cosh(z₁+ z₂) = Cosh(z₁)Cosh (z₂) + Sinh(z₁)Sinh(z₂)

Sinh(z +) = Sinh(z)

Cosh(z +) = Cosh(z)

Cosh(-z) = Cosh(z)

Sinh(-z) = -Sinh(z)

There is a connection between complex hyperbolic and complex trigonometric functions:

Cosh(z) = Cos(iz)

Sinh(z) = - iCos(iz)

Cos(z) = Cosh(iz)

Sin(z) = - iSinh(iz)

Sinh(iz) = iSin(z)

Sin(iz) = iSin(z)

by Tetyana Butler