arccoth
The inverse hyperbolic cotangent function.
Syntax

arccoth(x)

x
is a real or complex number

Description
arccoth
is the inverse of the hyperbolic cotangent function. For real numbers, the domain is ℝ ∖ [−1, 1]. It is defined by
arccoth(z) = ln((z + 1)/(z − 1)) / 2
for all complex z
. This formula is also used for real numbers in (−1, 1).
Notes
This function is also called arcoth
(area hyperbolic cotangent) in the literature. Some authors claim that the name arccoth
is a misnomer, but this depends crucially on how you view the connections between the trigonometric and the hyperbolic functions and their (restriction) inverses. One can argue that the name arccoth
makes perfect sense.
Examples
arccoth(e)
0.385968416453
arccoth(−π)
−0.329765314957
arccoth(0)
1.57079632679⋅i
arccoth(1/e)
0.385968416453 − 1.57079632679⋅i
arccoth(i)
−0.785398163397⋅i