The Art of Interface

# coth or cth — hyperbolic cotangent function

Category. Mathematics.

Abstract. Hyperbolic cotangent: definition, graph, properties and identities.

## 1. Definition

Hyperbolic cotangent is defined as

cothx ≡ (ex + ex) /(ex − ex)

## 2. Graph

Hyperbolic cotangent is antisymmetric function defined everywhere on real axis, except its singular point 0 — so, function domain is (−∞, 0)∪(0, +∞). Its graph is depicted below — fig. 1. Fig. 1. Graph of the hyperbolic cotangent function y = cothx.

Function codomain is entire real axis with gap in the middle: (−∞, −1)∪(1, +∞).

## 3. Identities

Base:

coth2x − csch2x = 1

By definition:

cothx ≡ coshx /sinhx ≡ 1 /tanhx

Property of antisymmetry:

coth−x = −cothx

Half-argument:

coth(x/2) = (1 + coshx) /sinhx
coth(x/2) = sinhx /(coshx − 1)
cothx = [1 + tanh2(x/2)] /[2 tanh(x/2)]

Doulbe argument:

coth(2x) = (coth2x + 1) /(2 cothx)

Triple argument:

coth(3x) = (coth3x + 3 cothx) /(3 coth2x + 1)

coth(4x) = (coth4x + 6 coth2x + 1) /(4 coth3x + 4 cothx + 1)

Power reduction:

coth2x = (cosh(2x) + 1) /(cosh(2x) − 1)
coth3x = (cosh(3x) + 3 coshx) /(sinh(3x) − 3 sinhx)
coth4x = (cosh(4x) + 4 cosh(2x) + 3) /(cosh(4x) − 4 cosh(2x) + 3)
coth5x = (cosh(5x) + 5 cosh(3x) + 10 coshx) /(sinh(5x) − 5 sinh(3x) + 10 sinhx)

Sum and difference of arguments:

coth(x + y) = (1 + cothx cothy) /(cothx + cothy)
coth(xy) = (1 − cothx cothy) /(cothx − cothy)

Product:

cothx cothy = [cosh(x + y) + cosh(xy)] /[cosh(x + y) − cosh(xy)]

Sum:

cothx + cothy = sinh(x + y) /(sinhx sinhy)
cothx − tanhy = sinh(yx) /(sinhx sinhy)

## 4. Support

Hyperbolic cotangent function coth or cth is supported in:

Hyperbolic cotangent function of the complex argument coth or cth is supported in:

## 5. How to use

To calculate hyperbolic cotangent of the number:

``coth(-1);``

To calculate hyperbolic cotangent of the current result:

``coth(Rslt);``

To calculate hyperbolic cotangent of the number x in memory:

``coth(Mem[x]);``