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Article 11 — Appendix A.19

coth or cth — hyperbolic cotangent function

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

1. Definition

Hyperbolic cotangent is defined as

cothx ≡ (ex + e−x) /(ex − e−x)

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 = coth x 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)

Quadruple argument:

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(x − y) = (1 − cothx cothy) /(cothx − cothy)

Product:

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

Sum:

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