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The inverse hyperbolic cotangent 
(Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic cotangent (Harris and Stocker 1998, p. 267),
is the multivalued function that is the inverse function of the hyperbolic
cotangent.
The variants 
and 
(Harris and Stocker 1998, p. 263) are sometimes
used to refer to explicit principal values of
the inverse hyperbolic cotangent, 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).
The function is sometimes denoted 
(Jeffrey 2000, p. 124) or 
(Gradshteyn and Ryzhik 2000, p. xxx). Note that
in the notation 
, 
is the hyperbolic tangent
and the superscript 
denotes an inverse function,
not the multiplicative inverse.
The principal value of 
is implemented in the Wolfram
Language as ArcCoth[z]
The inverse hyperbolic cotangent 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 ![[-1,1]](/images/equations/InverseHyperbolicCotangent/Inline12.svg)
. This follows from the definition of 
as
![coth^(-1)z=1/2[ln(1+1/z)-ln(1-1/z)].](/images/equations/InverseHyperbolicCotangent/NumberedEquation1.svg) |
(1)
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The inverse hyperbolic cotangent is given in terms of the inverse cotangent by
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(2)
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(Gradshteyn and Ryzhik 2000, p. xxx). For 
or 
, this simplifies to
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(3)
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The derivative is
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(4)
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and its indefinite integral is
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(5)
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It has the special values
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(6)
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(7)
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(8)
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(9)
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It has series expansions
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(10)
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(11)
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(12)
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(13)
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(OEIS A005408).
