Use coth for hyperbolic cotangent

coth(x) returns the hyperbolic cotangent, cosh(x) / sinh(x). Equivalently, it is 1 / tanh(x). It occurs in hyperbolic-function identities and models involving exponential growth and decay.

Behavior and relationships

For real nonzero x, coth(x) has magnitude greater than 1. It approaches 1 for large positive x and -1 for large negative x, but has a vertical singularity at zero. Compare tanh, which stays between -1 and 1 for real input, and use cosh and sinh to inspect the numerator and denominator.

Examples

Evaluate coth(1) for a positive real result a little above 1. In symbolic work, identities such as coth(x)^2 - csch(x)^2 = 1 can help simplify a formula; see csch for the reciprocal of sinh. The inverse hyperbolic cotangent is acoth, subject to its branch and domain conventions.

Input cautions

Never evaluate coth at zero, where division by sinh(0) occurs. Values close to zero may be enormous and magnify rounding error. For complex inputs, poles occur where sinh is zero, not merely at real zero. If an application only needs stable behavior far from zero, coth is straightforward; near a pole, reformulate or handle the singular case explicitly.