sech
coth(x)Input: x · Output: no output
What it is
The other three hyperbolics: coth(x) = cosh(x)/sinh(x), sech(x) = 1/cosh(x), csch(x) = 1/sinh(x). x is a real number or Complex. coth/csch have a pole at x=0 (and, for a Complex argument, at every x = i*k*pi) — evaluating there is an error, not a silent inf/NaN. sech has no real pole (cosh is never zero).
Overloads
One call shape — this function is not overloaded.
| Code | Input | Output | Description |
coth(x) | x number or Complexnot documented yet | not documented yet | sech has no real pole (cosh is never zero). |
Examples
No documentation example calls this form yet. The Core maths chapter is where one belongs.
How it works
coth/sech/csch: the other three hyperbolics, added 2026-09-16 alongside tanh -- coth is the closed form for a Warburg boundary condition tanh does not cover (EIS/BMS work, Ahmed's own domain), so shipping tanh alone left half the reason for fixing it undone. sech(x) = 1/cosh(x) has no real pole (cosh is never zero for real x); coth and csch both have one at x=0, and in the complex plane at every x = i*k*pi -- checked with map1_complex_checked rather than handed back as a silent inf, per Ahmed's instruction for this batch (a narrower rule than sqrt/ log's existing "NaN at a domain edge" convention: a pole is not merely out of domain, the function has no value there).
From the note above this builtin in the interpreter, where the reasoning was written down as it was built.