Series of hyperbolic sine and cosine · Mathematics

Definitions

The hyperbolic functions are built from the exponential function. They are called hyperbolic because (cosh t, sinh t) lies on the hyperbola x² − y² = 1, as (cos t, sin t) lies on the circle x² + y² = 1.

Definitions of sinh, cosh and tanh.

The basic identity and derivatives. Compare with the circular functions: there is no minus sign in the derivative of cosh.

An odd function rising steeply, like a stretched S.

An even function with minimum 1 at x = 0: the shape of a hanging chain (a catenary).

Logarithmic form of the inverse functions

Inverse hyperbolic functions.

Series

The Maclaurin series of the hyperbolic functions are sinh x = x + x³/3! + x⁵/5! + … and cosh x = 1 + x²/2! + x⁴/4! + …, with all terms positive.

Try itCheck the identity cosh² − sinh² = 1

Choose x. The identity holds for every value.

Check yourself sinh 0 =

Check yourself cosh² x − sinh² x =

Check yourself d/dx (sinh x) =

Key terms

Hyperbolic function: A function defined from eˣ and e⁻ˣ, such as sinh and cosh.
Catenary: The curve of a hanging chain, y = a cosh(x/a).
Inverse hyperbolic function: For example sinh⁻¹ x = ln(x + √(x² + 1)).
Even function: A function with f(−x) = f(x).

Summary

sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2.
cosh²x − sinh²x = 1.
The inverses have logarithmic forms.
The derivative of sinh is cosh, and of cosh is sinh.