csch2x

The hyperbolic cosecant function, denoted as csch(x) or csch2x, is a mathematical function defined as the reciprocal of the hyperbolic sine function. Specifically, csch(x) = 1 / sinh(x). The hyperbolic sine function, sinh(x), is defined in terms of the exponential function as sinh(x) = (e^x - e^-x) / 2. Therefore, csch(x) can also be expressed as csch(x) = 2 / (e^x - e^-x).

The domain of the csch(x) function is all real numbers except for x = 0, because sinh(0) = 0, and division by zero is undefined. The range of csch(x) is all real numbers except for the interval (-1, 1). The function has vertical asymptotes at x = 0, and horizontal asymptotes at y = 0 as x approaches positive or negative infinity.

In the context of the hyperbolic functions, csch(x) plays a role analogous to the cosecant function (csc(x)) in trigonometry. Its properties and behavior are closely related to those of other hyperbolic functions such as sinh(x), cosh(x), and tanh(x). The derivative of csch(x) is -csch(x)coth(x), and its integral is ln|csch(x) - coth(x)| + C. Hyperbolic functions, including csch(x), appear in solutions to certain differential equations, in the study of areas of hyperbolic sectors, and in various areas of physics and engineering.