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In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" (// or //), and the hyperbolic cosine "cosh" (//), from which are derived the hyperbolic tangent "tanh" (// or //), hyperbolic cosecant "csch" or "cosech" (// or //), hyperbolic secant "sech" (// or //), and hyperbolic cotangent "coth" (// or //), corresponding to the derived trigonometric 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. The hyperbolic functions take 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.
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.
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.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. The abbreviations sh and ch are still used in some other languages, like French and Russian.
The hyperbolic functions are:
Hyperbolic functions can be introduced via imaginary circular angles:
where i is the imaginary unit defined by i2 = −1.
Odd and even functions:
Hyperbolic sine and cosine satisfy the identity
which is similar to the Pythagorean trigonometric identity. One also has
for the other functions.
It can be shown that the area under the curve of cosh (x) over a finite interval is always equal to the arc length corresponding to that interval:
Sums of arguments:
Sum and difference of cosh and sinh:
The following integrals can be proved using hyperbolic substitution.
where C is the constant of integration.
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(−x), 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.
Since the area of a circular sector is it will be equal to u when r = square root of 2. In the diagram such a circle is tangent to the hyperbola x y = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the red augmentation depicts an area and magnitude as hyperbolic angle.
The legs of the two right triangles with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions.
Mellon Haskell of University of California, Berkeley described the basis of hyperbolic functions in areas of hyperbolic sectors in an 1895 article in Bulletin of the American Mathematical Society (see External links). He refers to the hyperbolic angle as an invariant measure with respect to the squeeze mapping just as circular angle is invariant under rotation.
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 argument formulas"
and the "half-argument 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 between two fixed points under uniform 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:
Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( for hyperbolic tangent and cotangent).
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