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 …

\displaystyle \text {tanh} (x + k\pi i) = \text {tanh}\ x tanh(x+kπi) = tanh x \displaystyle \text {coth} (x + k\pi i) =\text {coth} x coth(x+kπi) = cothx Relationship between inverse hyperbolic and …

5 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).

coth (z) = i cot (iz)

gives the hyperbolic cotangent of z. Mathematical function, suitable for both symbolic and numerical manipulation. automatically evaluates to exact values when its argument is the …

Here is a graphic of the hyperbolic cotangent function for real values of its argument . The hyperbolic cotangent function can be represented using more general mathematical functions.

Theorem Let z ∈ C z ∈ C be a complex number. Then: coth z = − cot(iz) coth ⁡ z = − cot ⁡ (i z) where: cot cot denotes the cotangent function coth coth denotes the hyperbolic cotangent i i is the imaginary unit: i2 = −1 i 2 = − 1. Proof

Coth [z] (840 formulas) Primary definition (1 formula) Specific values (88 formulas) General characteristics (9 formulas) Series representations (27 formulas) Integral representations (2 formulas) Limit representations (1 formula) Continued fraction representations (6 formulas) Differential equations (2 formulas) Transformations (91 formulas)

The real hyperbolic cotangent function is defined on the real numbers as: coth: R≠0 → R coth: R ≠ 0 → R: ∀x ∈ R≠0: coth x:= ex +e−x ex −e−x ∀ x ∈ R ≠ 0: coth ⁡ x:= e x + e − x e x − e − x where it is noted that at x = 0 x = 0: ex −e−x = 0 e x − e − x = …

The hyperbolic cotangent of z in Math. Defined by \ [ \coth z = \frac { \cosh z } { \sinh z } \] Real part on the real axis: