in the point, where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice 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. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used and to refer to circular functions and and to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to those used today. The abbreviations,,, are also currently used, depending on personal preference.
Definitions
There are various equivalent ways to define the hyperbolic functions.
The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the unique solution of the system such that and. They are also the unique solution of the equation, such that, for the hyperbolic cosine, and, for the hyperbolic sine.
Complex trigonometric definitions
Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:
Hyperbolic sine:
:
Hyperbolic cosine:
:
Hyperbolic tangent:
:
Hyperbolic cotangent:
:
Hyperbolic secant:
:
Hyperbolic cosecant:
:
where is the imaginary unit with. The above definitions are related to the exponential definitions via Euler's formula.
Characterizing properties
Hyperbolic cosine
It can be shown that the area under the curve of the hyperbolic cosine over a finite interval is always equal to the arc length corresponding to that interval:
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 for,, or and, 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 two sinhs. Odd and even functions: Hence: Thus, and are even functions; the others are odd functions. Hyperbolic sine and cosine satisfy: the last of which is similar to the Pythagorean trigonometric identity. One also has for the other functions.
The following integrals can be proved using hyperbolic substitution: where C is the constant of integration.
Taylor series expressions
It is possible to express the above functions as Taylor series: The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh, and sinh 0 = 0. The function cosh x has a Taylor series expression with only even exponents for x. Thus it is an even function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of the exponential function. where:
Comparison with circular functions
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle. Since the area of a circular sector with radius r and angle u is r^{2}u/2, it will be equal to u when r =. In the diagram such a circle is tangent to the hyperbola xy = 1 at. The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. The legs of the two right triangles with hypotenuse on the ray defining the angles are of length times the circular and hyperbolic functions. The hyperbolic angle is an invariant measure with respect to the squeeze mapping, just as the circular angle is invariant under rotation. 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 is the catenary, the curve formed by a uniform flexible chain hanging freely between two fixed points under uniform gravity.
Relationship to the exponential function
The decomposition of the exponential function in its even and odd parts gives the identities and The first one is analogous to Euler's formula Additionally,
Hyperbolic functions for complex numbers
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: so: Thus, hyperbolic functions are periodic with respect to the imaginary component, with period .