cschx

Cschx, more commonly written as csch(x) or csch x, is the hyperbolic cosecant function. It is defined as the reciprocal of the hyperbolic sine: csch x = 1 / sinh x, where sinh x = (e^x − e^−x)/2. For real arguments, the domain is all real numbers except x = 0, since sinh 0 = 0.

In the complex plane, csch z has simple poles at z = iπk for all integers k, with residues alternating as (−1)^k. The function is odd: csch(−x) = −csch(x). As x approaches zero, csch x behaves like 1/x (a Laurent-type singularity). As |x| grows large, csch x tends to 0.

Real-valued behavior: on the positive real axis, csch x is positive and decreases from +∞ to 0; on the negative real axis, it is negative and increases from −∞ to 0. Consequently, the range of csch x for real x is all nonzero real numbers.

Series and approximations: near x = 0, csch x has a Laurent expansion csch x = 1/x − x/6 + 7x^3/360 − 31x^5/15120 + … . This yields practical approximations for small arguments.

Relation to other functions: csch x is the reciprocal of sinh x, so many identities follow from that relationship. It is connected to the ordinary cosecant by analytic continuation: csch x = − i csc(i x).

Applications and context: csch x appears in various areas of mathematics, physics, and engineering, including solutions to certain differential equations, hyperbolic geometry, and integrals involving hyperbolic functions. See also the hyperbolic functions and the ordinary cosecant for related concepts.