Arccothx

The inverse hyperbolic cotangent function, denoted as arccoth(x) or coth⁻¹(x), is a special function in mathematics that represents the inverse of the hyperbolic cotangent function. The hyperbolic cotangent, coth(x), is defined as the ratio of the hyperbolic cosine to the hyperbolic sine, expressed as coth(x) = cosh(x)/sinh(x). Since coth(x) is not one-to-one over its entire domain, its inverse is typically defined by restricting the domain to positive real numbers greater than 1 or negative real numbers less than -1.

The arccoth(x) function is defined for all real numbers x with |x| > 1. For x > 1, the function can be expressed in terms of logarithms as arccoth(x) = (1/2) * ln((x + 1)/(x - 1)). For x < -1, the result is equivalent to π + (1/2) * ln((x + 1)/(x - 1)), accounting for the branch cut and ensuring continuity. The function is odd, meaning arccoth(-x) = -arccoth(x), and it is strictly decreasing on its domain.

Arccoth(x) has applications in various fields, including complex analysis, differential equations, and integral calculus. It is also used in the study of hyperbolic geometry and special functions. The derivative of arccoth(x) is given by 1/(x² - 1), and its integral can be expressed in terms of logarithmic functions. Additionally, arccoth(x) can be related to other inverse hyperbolic functions, such as arctanh(x), through identities involving logarithms and algebraic manipulations.