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 …

Apr 17, 2019 · I am trying to solve $\sinh z = i$; this has to result in $2\pi n + (1/2) \pi i$. When applying the exponential form of $$\sinh z = \frac {e ^ z - e ^ {-z}} 2$$ and later the quadratic …

Apr 30, 2025 · 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 …

Here is a graphic of the hyperbolic sine function for real values of its argument . The function is a particular case of more complicated mathematical functions. For example, it is a special case …

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 …

cosh(z) = cos(iz) and sinh(z) = −isin(iz). It follows that sinh(−z) = −isin(−iz) = isin(iz) = −sinh(z), cosh(−z) = cos(−iz) = cos(iz) = cosh(z), cosh2(z)−sinh2(z) = cos2(iz)−(−isin(iz))2 = …

cosh 2 (x) - sinh 2 (x) = 1 tanh 2 (x) + sech 2 (x) = 1 coth 2 (x) - csch 2 (x) = 1 Inverse Hyperbolic Defintions. arcsinh(z) = ln( z + (z 2 + 1) ) arccosh(z) = ln( z (z 2 - 1) ) arctanh(z) = 1/2 ln( …

This sinh calculator allows you to quickly determine the values of the hyperbolic sine function.