Cothx

Cothx is a term used to denote the hyperbolic cotangent of the real variable x. In standard mathematical notation it is written as coth(x) or coth x, and in some programming or compact-notation contexts it appears as a single token, cothx.

For x ≠ 0, cothx = cosh x / sinh x. The function is undefined at x = 0 and has a simple pole there. Cothx is an odd function, since cosh x is even and sinh x is odd, giving coth(-x) = -coth(x). It is also related to the reciprocal of the hyperbolic tangent, since cothx = 1 / tanh x.

The derivative of cothx is d/dx cothx = -csch^2 x, where csch x = 1 / sinh x. The Laurent series around x = 0 is cothx = 1/x + x/3 - x^3/45 + 2x^5/945 - ..., reflecting its simple pole at the origin. As x grows large in magnitude, cothx approaches 1 in magnitude; for positive x, a common asymptotic form is cothx = 1 + 2/(e^{2x} - 1), illustrating the rapid decay of the excess over 1.

Cothx appears in solutions to differential equations involving hyperbolic functions and in various physical and geometric contexts where rapidity or hyperbolic geometry is relevant. It also arises in integral evaluations and in the study of series and asymptotics involving hyperbolic functions.

Notes: the notation cothx is simply a concatenated variant of coth(x) used in some texts and programming environments. Related functions include tanh, coth, sech, and csch.