Hyperbolic Functions

Hyperbolic function is an analog of the ordinary trigonometric function, also called circular function. The basic hyperbolic functions are the hyperbolic sine "$\sinh$," and the hyperbolic cosine "$\cosh$," from which are derived the hyperbolic tangent "$\tanh$," and so on, corresponding to the derived functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh" or sometimes "arcsinh") and so on.

Just as the points ($\cos t$, $\sin t$) form a circle with a unit radius, the points ($\cosh t$, $\sinh t$) form the right half of the equilateral hyperbola. Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, of some cubic equations, and of Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take real values for a real argument called a hyperbolic angle. The size of a hyperbolic angle is the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence meromorphic.

The hyperbolic functions are as follows.

Hyperbolic sine

$\sinh (x) = \dfrac{e^x - e^{-x}}{2}$

Hyperbolic cosine

$\cosh (x) =\dfrac{e^x + e^{-x}}{2}$

Hyperbolic tangent

$\tanh (x) = \dfrac{\sinh(x)}{\cosh (x)} = \dfrac{1 - e^{-2x}}{1 + e^{-2x}}$

Hyperbolic cotangent

$\coth (x) = \dfrac{\cosh (x)}{\sinh (x)} = \dfrac{1 + e^{-2x}}{1 - e^{-2x}}$

Hyperbolic secant

$\text{sech} (x) = (\cosh (x))^{-1}= \dfrac{2}{e^x + e^{-x}}$

Hyperbolic cosecant

$\text{csch} (x)= (\sinh (x))^{-1}= \dfrac{2}{e^x - e^{-x}}$