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
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## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
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## Under source code control:   1997/09/06 20:03:34
## File existed as early as:    1997
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## chongo <was here> /\oo/\     http://www.isthe.com/chongo/
## Share and enjoy!  :-)        http://www.isthe.com/chongo/tech/comp/calc/
