$\text{tanh} (x + k\pi i) = \text{tanh}\ x$ $\text{coth} (x + k\pi i) =\text{coth} x$ Relationship between inverse hyperbolic and inverse trigonometric functions

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 with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

COTH (x) returns the hyperbolic cotangent of x. The argument x can be a real number or a matrix. When it is a matrix, the function returns a matrix with the same dimensions and with the COTH function applied to all elements.

Graphs of Hyperbolic Functions. The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are presented. The six hyperbolic functions are defined as follows: Hyperbolic Sine Function : \( \sinh(x) = \dfrac{e^x - e^{-x}}{2} \)

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Nov 25, 2024 · The hyperbolic cotangent function (coth) is the reciprocal of the hyperbolic tangent function, which is given by the expression: coth x = ${\dfrac{1}{\tanh x}}$ = ${\dfrac{e^{x}+e^{-x}}{e^{x}-e^{-x}}}$

Feb 6, 2021 · Here is a graph of the function: coth as cosh divided by sinh. The coth function involves two functions, $\cosh{x}$ divided by $\sinh{x}$. This animation illustrates this: When $x$ is large and negative, $\sinh{x}$ becomes ever closer to $-\cosh{x}$. The value of $\coth{x}$ therefore tends towards -1.

4 days ago · The hyperbolic cotangent is defined as cothz= (e^z+e^ (-z))/ (e^z-e^ (-z))= (e^ (2z)+1)/ (e^ (2z)-1). (1) The notation cthz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Coth [z].

Hyperbolic cotangent is antisymmetric function defined everywhere on real axis, except its singular point 0 — so, function domain is (−∞, 0)∪ (0, +∞). Its plot is depicted below — fig. 1. Fig. 1. Plot of the hyperbolic cotangent function y = cothx. Function codomain is entire real axis with gap in the middle: (−∞, −1)∪ (1, +∞). 3. Identities. 4.

Mathematically, coth (x) can be expressed as: coth (x) = cosh (x) / sinh (x) Here, cosh (x) represents the hyperbolic cosine function and sinh (x) represents the hyperbolic sine function. To understand the behavior of coth (x), it is helpful to consider its graph.