Package or command for inverse hyperbolic trig functions?

Question

What are the commands for all the:

• inverse trig functions (arcsec(x), arcCsc(x), arcCot(x));
• hyperbolic trig functions (sech(x) and csch(x)), and
• inverse hyperbolic trig functions?

Answer 1

After spending some time looking for this, I found this post that suggested defining the new commands for the omitted inverse trig functions.

Here I've augmented that with the full suit of hyperbolic and inverse hyperbolic functions for convenience, as google doesn't turn anything up for this search, nor does the other post come up if one is searching for the inverse hyperbolic functions, specifically.

\documentclass{article} 

\usepackage{amsmath}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\arcsec}{arcsec}
\DeclareMathOperator{\arccot}{arcCot}
\DeclareMathOperator{\arccsc}{arcCsc}
\DeclareMathOperator{\arccosh}{arcCosh}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsech}{arcsech}
\DeclareMathOperator{\arccsch}{arcCsch}
\DeclareMathOperator{\arccoth}{arcCoth} 

\begin{document} 
\[
\sech x   \cschx   \arcsec x   \arccot x   \arccsc x   \arccosh x   \arcsinh x   \arctanh x   \arcsech x   arccsch x   \arccoth x 
\]
\end{document}

Answer 2

A bit more about "c or not to c"...

The inverse hyperbolic sine sinh^(-1) z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) and sometimes denoted arcsinh z (Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic sine. The variants Arcsinh z or Arsinh z (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic sine, although this distinction is not always made. Worse yet, the notation arcsinh z is sometimes used for the principal value, with Arcsinh z being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation sinh^(-1) z, sinh z is the hyperbolic sine and the superscript -1 denotes an inverse function, not the multiplicative inverse.

Its principal value of sinh^(-1) z is implemented in Mathematica as ArcSinh[z] and in the GNU C library as asinh(double x).

The above is from Wolfram MathWorld.

In summary, there are numerous notations for these functions, and ultimately it is the publisher's choice.