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sinh, cosh and tanh
csch, sech and coth
Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the equilateral hyperbola x² - y² = 1. This is based on the easily verified identity
However, the hyperbolic functions are not periodic.
The parameter t is not a circular angle, but rather an 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, and cosh 0 = 1
The function sinh x is an odd function, that is symmetric with respect to the origin, and sinh 0 = 0
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborne'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 two sinh's. This yields for example the addition theorems
and the "half-angle formulas"
The derivative of sinh x is given by cosh x and the derivative of cosh x is sinh x.
The graph of the function cosh x is the catenary curve.
The inverses of the hyperbolic functions are
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; their Taylor series expansions are given in the Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point article.
Relationships to regular trigonometric functions are given by Euler's_formulaThis article is about the Euler's formula in complex analysis . There is also Euler's formula which is related to the Euler characteristic in algebraic topology. Euler's formula named after Leonhard Euler, is a mathematical formula in the subfield of comp for complex numbers:
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