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NAME
gd - Gudermannian function
SYNOPSIS
gd(z [,eps])
TYPES
z number (real or complex)
eps 0 < real < 1, defaults to epsilon()
return number or "Log of zero or infinity" error value
DESCRIPTION
Calculate the Gudermannian of z to a multiple of eps with errors in
real and imaginary parts less in absolute value than .75 * eps,
or return an error value if z is close to one of the branch points
at odd multiples of (pi/2) * i.
gd(z) is usually defined initially for real z by one of the formulae
gd(z) = 2 * atan(exp(z)) - pi/2
= 2 * atan(tanh(z/2))
= atan(sinh(z)),
or as the integral from 0 to z of (1/cosh(t))dt. For complex z, the
principal branch, approximated by gd(z, eps), has the cut:
re(z) = 0, abs(im(z)) >= pi/2; on the cut calc takes gd(z) to be
the limit as z is approached from the right or left according as
im(z) > or < 0.
If z = x + y*i and abs(y) < pi/2, gd(z) is given by
gd(z) = atan(sinh(x)/cos(y)) + i * atanh(sin(y)/cosh(x)).
EXAMPLE
; print gd(1, 1e-5), gd(1, 1e-10), gd(1, 1e-15)
0.86577 0.8657694832 0.865769483239659
; print gd(2+1i, 1e-5), gd(2+1i, 1e-10)
1.42291+0.22751i 1.4229114625+0.2275106584i
LIMITS
0 < eps < 1
LINK LIBRARY
COMPLEX *c_gd(COMPLEX *x, NUMBER *eps)
SEE ALSO
agd, exp, ln, sin, sinh, etc.
## Copyright (C) 1999,2021,2023 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 20:03:35
## 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/
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