acoth(): The Inverse Hyperbolic Cotangent

acoth() is the inverse of coth() on its principal branch. Hyperbolic cotangent occurs in exponential models, boundary-value problems, and some statistical and physical formulas. acoth converts an eligible hyperbolic ratio back into its corresponding parameter.

Domain and numerical caution

For real values, acoth(x) is defined when x is less than −1 or greater than 1. It is not real on the interval from −1 through 1, and it becomes singular at −1 and 1. The identity acoth(x) = 0.5 × log((x + 1)/(x − 1)) is useful for understanding both restrictions and sensitivity near the boundaries.

Before evaluating a data-derived ratio, test its magnitude with abs(). Values very close to ±1 can produce large outputs, so retain sufficient precision and avoid rounding input too early. If complex results are expected, make that choice deliberately rather than treating a real-domain failure as a formatting issue.

Related functions

acoth belongs with asinh(), acosh(), and atanh(), but their real domains are not interchangeable. You can check a valid result by computing coth(acoth(x)).

Hyperbolic functions are not angle functions in the ordinary geometric sense. Their arguments are dimensionless parameters, commonly derived from exponentials, which is why consistent units and scaling still matter before using acoth.