NAME
    agd - inverse Gudermannian function

SYNOPSIS
    agd(z [,eps])

TYPES
    z		number (real or complex)
    eps		nonzero real, defaults to epsilon()

    return	number or infinite error value

DESCRIPTION
    Calculate the inverse Gudermannian of z to a multiple of eps with
    errors in real and imaginary parts less in absolute value than .75 * eps,
    or an error value if z is very close to one of the one of the branch
    points of agd(z)..

    agd(z) is usually defined initially for real z with abs(z) < pi/2 by
    one of the formulae

		 agd(z) = ln(sec(z) + tan(z))

			= 2 * atanh(tan(z/2))

			= asinh(tan(z)),

    or as the integral from 0 to z of (1/cos(t))dt.  For complex z, the
    principal branch, approximated by gd(z, eps), has cuts along the real
    axis outside -pi/2 < z < pi/2.

    If z = x + i * y and abs(x) < pi/2, agd(z) is given by

	agd(z) = atanh(sin(x)/cosh(y)) + i * atan(sinh(y)/cos(x)>

EXAMPLE
    ; print agd(1, 1e-5), agd(1, 1e-10), agd(1, 1e-15)
    1.22619 1.2261911709 1.226191170883517

    ; print agd(2, 1e-5), agd(2, 1e-10)
    1.52345-3.14159i 1.5234524436-3.1415926536i

    ; print agd(5, 1e-5), agd(5, 1e-10), agd(5, 1e-15)
    -1.93237 -1.9323667197 -1.932366719745925

    ; print agd(1+2i, 1e-5), agd(1+2i, 1e-10)
    0.22751+1.42291i 0.2275106584+1.4229114625i

LIMITS
    none

LINK LIBRARY
    COMPLEX *c_agd(COMPLEX *x, NUMBER *eps)

SEE ALSO
    gd, exp, ln, sin, sinh, etc.

## Copyright (C) 1999,2021  Landon Curt Noll
##
## Calc is open software; you can redistribute it and/or modify it under
## the terms of the version 2.1 of the GNU Lesser General Public License
## as published by the Free Software Foundation.
##
## Calc is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
## Public License for more details.
##
## A copy of version 2.1 of the GNU Lesser General Public License is
## distributed with calc under the filename COPYING-LGPL.  You should have
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
##
## Under source code control:	1997/09/06 20:03:34
## File existed as early as:	1997
##
## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
