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The inverse hyperbolic cosecant 
(Zwillinger 1995, p. 481), sometimes called
the area hyperbolic cosecant (Harris and Stocker 1998, p. 271) and sometimes
denoted 
(Beyer 1987, p. 181) or 
(Abramowitz and Stegun 1972, p. 87; Jeffrey 2000,
p. 124), is the multivalued function
that is the inverse function of the hyperbolic
cosecant. The variants 
(Abramowitz and Stegun 1972, p. 87) and 
(Harris and Stocker 1998, p. 263) are sometimes
used to refer to explicit principal values. 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 cosecant
and the superscript 
denotes an inverse function,
not the multiplicative inverse.
The inverse hyperbolic cosecant is a multivalued function and hence requires a branch cut in the
complex plane, which the Wolfram
Language's convention places at 
.
The principal value of 
is implemented in the Wolfram
Language as ArcCsch[z].
It has special value
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(1)
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where 
is the golden ratio.
The inverse hyperbolic cosecant is a multivalued function and hence requires a branch cut in the
complex plane, which the Wolfram
Language's convention places at the line segment 
. This follows from the definition of 
as
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(2)
|
The derivative of the inverse hyperbolic cosecant is
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(3)
|
and the indefinite integral is
![intcsch^(-1)zdz
=zcsch^(-1)z+ln[z(1+sqrt((1+z^2)/(z^2)))]+C.](/images/equations/InverseHyperbolicCosecant/NumberedEquation4.svg) |
(4)
|
For real 
,
it satisfies
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(5)
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The inverse hyperbolic cosecant has Puiseux series
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(6)
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(7)
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(OEIS A052468 and A052469) about 0, and Taylor series about 
of
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(8)
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(9)
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(10)
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(OEIS A055786 and A002595), where 
is a Legendre polynomial and 
is a Pochhammer symbol.
