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Since the function cosh x is even, only even exponents for x occur in its Taylor series. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.
Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
The hyperbolic functions sinh, cosh, and tanh can be illustrated with hyperbolic coordinates. Let ... that as the abscissas increased in a geometric series, the sum ...
The even and odd terms of this series provide sums denoting cosh(x) and sinh(x), so that = + . These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing (−1) k into the series, resulting in alternating series.
Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and ...