csch

csch, short for hyperbolic cosecant, is a hyperbolic function defined by csch x = 1 / sinh x for real x. It can also be written as csch x = 2 / (e^x − e^−x). On the real line, sinh x vanishes only at x = 0, so csch x is defined for all real x ≠ 0 and has a simple pole at x = 0. The function is odd, since sinh is odd, so csch(−x) = −csch(x). Its range is all real numbers except zero.

Key relationships and properties include the identity coth^2 x − csch^2 x = 1, which implies coth^2 x = 1 + csch^2 x. The derivative is d/dx csch x = − csch x coth x, making it useful in various calculus contexts. In terms of asymptotics, as |x| → ∞, csch x decays exponentially: |csch x| ~ 2 e^{−|x|} (positive for x > 0, negative for x < 0).

Series and values near zero: since sinh x ~ x + x^3/6 + ..., one has csch x ~ 1/x − x/6 + 7x^3/360 − 31x^5/15120 + …, indicating a simple pole at the origin and a rapid approach to zero as |x| grows.

Complex extension: csch z is defined by csch z = 1 / sinh z for complex z and is a meromorphic function with simple poles at z = nπ i (n ∈ Z). It does not vanish for any real argument, and it shares the general behavior of reciprocal hyperbolic functions in mathematical analysis and physics.

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