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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.
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The Poincaré half-plane model is closely related to a model of the hyperbolic plane in the quadrant Q = {(x,y): x > 0, y > 0}. For such a point the geometric mean = and the hyperbolic angle = / produce a point (u,v) in the upper half-plane.
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 ...
therefore the explicit form of the boost matrix depends only the generator and its square. Splitting the power series into an odd power series and an even power series, using the odd and even powers of the generator, and the Taylor series of sinh ϕ and cosh ϕ about ϕ = 0 obtains a more compact but detailed form of the boost matrix
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Rapidity is the parameter expressing variability of an event on the hyperbola which represents the future events one time unit away from the origin O. These events can be expressed (sinh w, cosh w) where sinh is the hyperbolic sine and cosh is the hyperbolic cosine.