### 5.8 Inverse Complex Hyperbolic Functions

Function: gsl_complex gsl_complex_arcsinh (gsl_complex z)

This function returns the complex hyperbolic arcsine of the complex number z, \arcsinh(z). The branch cuts are on the imaginary axis, below -i and above i.

Function: gsl_complex gsl_complex_arccosh (gsl_complex z)

This function returns the complex hyperbolic arccosine of the complex number z, \arccosh(z). The branch cut is on the real axis, less than 1. Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving \arccosh(z)=\log(z-\sqrt{z^2-1}).

Function: gsl_complex gsl_complex_arccosh_real (double z)

This function returns the complex hyperbolic arccosine of the real number z, \arccosh(z).

Function: gsl_complex gsl_complex_arctanh (gsl_complex z)

This function returns the complex hyperbolic arctangent of the complex number z, \arctanh(z). The branch cuts are on the real axis, less than -1 and greater than 1.

Function: gsl_complex gsl_complex_arctanh_real (double z)

This function returns the complex hyperbolic arctangent of the real number z, \arctanh(z).

Function: gsl_complex gsl_complex_arcsech (gsl_complex z)

This function returns the complex hyperbolic arcsecant of the complex number z, \arcsech(z) = \arccosh(1/z).

Function: gsl_complex gsl_complex_arccsch (gsl_complex z)

This function returns the complex hyperbolic arccosecant of the complex number z, \arccsch(z) = \arcsin(1/z).

Function: gsl_complex gsl_complex_arccoth (gsl_complex z)

This function returns the complex hyperbolic arccotangent of the complex number z, \arccoth(z) = \arctanh(1/z).