arcsech
The inverse hyperbolic secant function.
Syntax
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arcsech(x)-
xis a real or complex number
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Description
arcsech is the inverse of the restriction of the hyperbolic secant function to [0, ∞). For real numbers, the domain is (0, 1]. It is defined by
arcsech(z) = ln(1/z + √(1/z + 1) ⋅ √(1/z − 1))
for all complex z. This formula is also used for real numbers outside [0, 1].
Notes
This function is also called arsech (area hyperbolic secant) in the literature. Some authors claim that the name arcsech 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 arcsech makes perfect sense.
Examples
arcsech(1/e)
1.65745445415
arcsech(1)
0
arcsech(2)
1.0471975512⋅i
arcsech(−1)
3.14159265359⋅i
arcsech(i)
0.88137358702 − 1.57079632679⋅i