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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. [12]
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous ...
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n). [1] All elementary functions are ...
Then the area of the hyperbolic sector is the area of the triangle minus the curved region past the vertex at (,): = = ( (+)), which simplifies to the area hyperbolic cosine = = (+). Solving for x {\displaystyle x} yields the exponential form of the hyperbolic cosine: x = cosh a = e a + e − a 2 . {\displaystyle x=\cosh a={\frac ...
Study of hyperbolic triangles in hyperbolic geometry with hyperbolic functions. Hyperbolic functions in Euclidean geometry: The unit circle is parameterized by (cos t, sin t) whereas the equilateral hyperbola is parameterized by (cosh t, sinh t). Gyrotrigonometry: A form of trigonometry used in the gyrovector space approach to hyperbolic ...
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