arctanh
The inverse hyperbolic tangent function.
Syntax
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arctanh(x)-
xis a real or complex number
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Description
arctanh is the inverse of the hyperbolic tangent function. For real numbers, the domain is (−1, 1). It is defined by
arctanh(z) = [ln(1 + z) − ln(1 − z)] / 2
for all complex z. This formula is also used for real numbers outside [−1, 1].
Notes
This function is also called artanh (area hyperbolic cotangent) in the literature. Some authors claim that the name arctanh 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 arctanh makes perfect sense.
Examples
arctanh(0)
0
arctanh(1/π)
0.329765314957
arctanh(2)
0.549306144334 − 1.57079632679⋅i
arctanh(i)
0.785398163397⋅i