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 nultiple 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)
    .22751+1.42291i .2275106584+1.4229114625i

LIMITS
    none

LIBRARY
    COMPLEX *cagd(COMPLEX *x, NUMBER *eps)

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