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The inverse hyperbolic cosine 
(Beyer 1987, p. 181; Zwillinger 1995, p. 481),
sometimes called the area hyperbolic cosine (Harris and Stocker 1998, p. 264)
is the multivalued function that is the inverse function of the hyperbolic
cosine.
The variants 
and 
(Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse 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 
(Abramowitz and Stegun 1972, p. 87; Jeffrey 2000,
p. 124) or 
(Gradshteyn and Ryzhik 2000, p. xxx). Note that in
the notation 
,
is the hyperbolic cosine and the superscript
denotes an inverse function, not the multiplicative
inverse.
The principal value of 
is implemented in the Wolfram
Language as ArcCosh[z],
and in the GNU C library as acosh(double x).
The inverse hyperbolic cosine 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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Gradshteyn and Ryzhik (2000, p. xxx) give a version of the inverse hyperbolic cosine which holds only in the upper half of the complex plane ![I[z]>0](/images/equations/InverseHyperbolicCosine/Inline14.svg)
and for 
. The corresponding corrected formulas are
![cosh^(-1)z={icos^(-1)z for 0<arg(z)<=pi or 0<z<1; -icos^(-1)z for I[z]<0 or z>1,](/images/equations/InverseHyperbolicCosine/NumberedEquation2.svg) |
(2)
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which can be written in general form as
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(Wolfram Functions Site).
The derivative of the inverse hyperbolic cosine is
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and its indefinite integral is
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(5)
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For real 
,
it satisfies
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The inverse hyperbolic cosine has the Maclaurin series,
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(7)
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(OEIS A055786 and A002595), where 
is a Pochhammer symbol.
![cosh^(-1)x=sqrt(2(x-1))[1-1/(12)(x-1)+3/(160)(x-1)^2-5/(896)(x-1)^3+...]](/images/equations/InverseHyperbolicCosine/NumberedEquation7.svg) |
(9)
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(OEIS A055786 and A091019) about 1, and the Taylor series
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(11)
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