In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" (typically pronounced /ˈsɪntʃ/ or /ˈʃaɪn/), and the hyperbolic cosine "cosh" (typically pronounced /ˈkɒʃ/), from which are derived the hyperbolic tangent "tanh" (typically pronounced /ˈtæntʃ/ or /ˈθæn/), etc., in analogy to the derived trigonometric functions. The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called "asinh", or sometimes by the misnomer of "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, and 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.
Hyperbolic functions were introduced in the 18th century by the Swiss mathematician Johann Heinrich Lambert.
The hyperbolic functions are:
Via complex numbers the hyperbolic functions are related to the circular functions as follows:
where is the imaginary unit defined as .
Note that, by convention, sinh2x means (sinhx)2, not sinh(sinhx); similarly for the other hyperbolic functions when used with positive exponents. Another notation for the hyperbolic cotangent function is , though cothx is far more common.
Hyperbolic sine and cosine satisfy the identity
which is similar to the Pythagorean trigonometric identity.
It can also be shown that the area under the graph of cosh x from A to B is equal to the arc length of cosh x from A to B.
For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions
In the above expressions, C is called the constant of integration.
It is possible to express the above functions as Taylor series:
A point on the hyperbola xy = 1 with x > 1 determines a hyperbolic triangle in which the side adjacent to the hyperbolic angle is associated with cosh while the side opposite is associated with sinh. However, since the point (1,1) on this hyperbola is a distance √2 from the origin, the normalization constant 1/√2 is necessary to define cosh and sinh by the lengths of the sides of the hyperbolic triangle.
and the property that cosh t ≥ 1 for all t.
The hyperbolic functions are periodic with complex period 2πi (πi for hyperbolic tangent and cotangent).
The parameter t is not a circular angle, but rather a hyperbolic angle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola.
The function cosh x is an even function, that is symmetric with respect to the y-axis.
The function sinh x is an odd function, that is −sinh x = sinh(−x), and sinh 0 = 0.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule  states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems
the "double angle formulas"
and the "half-angle formulas"
The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin x).
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.
The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain hanging freely under gravity.
From the definitions of the hyperbolic sine and cosine, we can derive the following identities:
These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums of complex exponentials.
Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic.
Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers: