Calculate hyperbolic cosecant with csch

csch(x) returns the hyperbolic cosecant, the reciprocal of hyperbolic sine: 1 / sinh(x). It is useful in identities and solutions involving hyperbolic functions, exponential forms, and differential equations.

Key behavior

For real x, csch is an odd function: changing the sign of x changes the sign of the result. Its magnitude becomes small as |x| grows, but it is undefined at zero because sinh is zero there. Compare it with sech, the reciprocal of cosh, which is defined at zero.

Identity and example

At x = 1, csch(1) is the reciprocal of sinh(1). A useful identity is coth(x)^2 - csch(x)^2 = 1; see coth for the other term. The inverse function acsch can recover a value subject to its principal branch.

Numerical cautions

Do not evaluate csch at zero, and treat values close to zero carefully because the reciprocal can be very large. For large magnitudes, exponential underflow or loss of precision may matter in an implementation. Use exp when an explicitly scaled exponential expression is more stable for your model.