Hyperbolic functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and , the derivatives of and are and . Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace 's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: from which are derived: corresponding to the derived trigonometric functions. The inverse hyperbolic functions are: 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 ("sinus/cosinus circulare") to refer to circular functions and and ("sinus/cosinus hyperbolico") 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. Exponential definitions. In terms of the exponential function: Differential equation definitions. The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the unique solution of the system formula_7 with the initial conditions formula_8 preventing any pair of functions formula_9 to be a solution. 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: where is the imaginary unit with . The above definitions are related to the exponential  definitions via Euler's formula (See below). 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: formula_16 Hyperbolic tangent. The hyperbolic tangent is the (unique) solution to the differential equation , with . Useful relations. 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 formula_17, formula_18, formula_19 or formula_17 and formula_21 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 containing a product of two sinhs. Odd and even functions: formula_22 Hence: formula_23 Thus, and are even functions; the others are odd functions. formula_24 Hyperbolic sine and cosine satisfy: formula_25 the last of which is similar to the Pythagorean trigonometric identity. One also has formula_26 for the other functions. Sums of arguments. formula_27 particularly formula_28 Also: formula_29 Subtraction formulas. formula_30 Also: formula_31 Half argument formulas. formula_32 where is the sign function. If , then formula_33  Square formulas. formula_34 Inequalities. The following inequality is useful in statistics: formula_35 It can be proved by comparing term by term the Taylor series of the two functions. Inverse functions as logarithms. formula_36 Derivatives. formula_37