In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle …

If x = sinh y, then y = sinh-1 a is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as …

3 days ago · The hyperbolic sine is defined as sinhz=1/2(e^z-e^(-z)). (1) The notation shz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram …

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use …

Also note that it follows that both sinh(z) and cosh(z) are periodic with period 2πi, that sinh(z) = 0 if and only if z = nπi, n = 0,±1,±2,..., and cosh(z) = 0 if and only if z = i

Feb 8, 2024 · Let $z \in \C$ be a complex number. Then: $i \sinh z = \map \sin {i z}$ where: $\sin$ denotes the complex sine $\sinh$ denotes the hyperbolic sine $i$ is the imaginary unit: $i^2 = …

Jan 15, 2020 · For any z ∈ C define the hyperbolic cosine and hyperbolic sine as: coshz = e z+e −z 2 and sinhz = e −e 2. Note 3.35.A. Since ez is an entire function, then coshz and sinhz are …

Sinh. Elementary Functions Sinh: Introduction to the Hyperbolic Sine Function : Defining the hyperbolic sine function : A quick look at the hyperbolic sine function : Representation through …

i sin(iz))2 = cos2(iz) + sin2(iz) = 1. cosh(z) = 0 if and only if z = i + n , n = 0; 1; 2; : : :. Note: The above are all analytic in their domains of de nition. csch(z) coth(z).

In the following description, \(z\) stands for the complex number. \(x\) stands for the real value \(Re\) and \(y\) for the imaginary value \(Im\). \(sinh(z) = sinh(x) · cos(y) + cosh(x) · sin(y)\)