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The inverse hyperbolic secant 
(Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic secant (Harris and Stocker 1998, p. 271)
and sometimes also denoted 
(Jeffrey 2000, p. 124), is the multivalued
function that is the inverse function of
the hyperbolic secant. The variants 
or 
(Harris and Stocker 1998, p. 263) are sometimes
used to refer to explicit principal values of
the inverse hyperbolic secant, although this distinction is not always made. Worse
yet, the notation 
is sometimes used for the principal value, with 
being used for the multivalued function (Abramowitz and Stegun 1972, p. 87).
Note that in the notation 
, 
is the hyperbolic secant
and the superscript 
denotes an inverse function,
not the multiplicative inverse.
The principal value of 
is implemented in the Wolfram
Language as ArcSech[z].
The inverse hyperbolic secant is a multivalued function and hence requires a branch cut in the
complex plane, which the Wolfram
Language's convention places at the line segments ![(-infty,0]](/images/equations/InverseHyperbolicSecant/Inline11.svg)
and 
. This follows from the definition of 
as
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(1)
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For real 
,
it satisfies
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(2)
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The derivative of the inverse hyperbolic secant is given by
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(3)
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and its indefinite integral is
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(4)
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It has Maclaurin series
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(5)
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(6)
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