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84 lines
2.5 KiB
Plaintext
84 lines
2.5 KiB
Plaintext
NAME
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agd - inverse gudermannian function
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SYNOPSIS
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agd(z [,eps])
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TYPES
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z number (real or complex)
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eps nonzero real, defaults to epsilon()
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return number or infinite error value
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DESCRIPTION
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Calculate the inverse gudermannian of z to a nultiple of eps with
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errors in real and imaginary parts less in absolute value than .75 * eps,
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or an error value if z is very close to one of the one of the branch
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points of agd(z)..
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agd(z) is usually defined initially for real z with abs(z) < pi/2 by
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one of the formulae
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agd(z) = ln(sec(z) + tan(z))
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= 2 * atanh(tan(z/2))
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= asinh(tan(z)),
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or as the integral from 0 to z of (1/cos(t))dt. For complex z, the
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principal branch, approximated by gd(z, eps), has cuts along the real
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axis outside -pi/2 < z < pi/2.
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If z = x + i * y and abs(x) < pi/2, agd(z) is given by
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agd(z) = atanh(sin(x)/cosh(y)) + i * atan(sinh(y)/cos(x)>
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EXAMPLE
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> print agd(1, 1e-5), agd(1, 1e-10), agd(1, 1e-15)
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1.22619 1.2261911709 1.226191170883517
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> print agd(2, 1e-5), agd(2, 1e-10)
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1.52345-3.14159i 1.5234524436-3.1415926536i
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> print agd(5, 1e-5), agd(5, 1e-10), agd(5, 1e-15)
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-1.93237 -1.9323667197 -1.932366719745925
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> print agd(1+2i, 1e-5), agd(1+2i, 1e-10)
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.22751+1.42291i .2275106584+1.4229114625i
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LIMITS
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none
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LINK LIBRARY
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COMPLEX *cagd(COMPLEX *x, NUMBER *eps)
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SEE ALSO
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gd, exp, ln, sin, sinh, etc.
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## Copyright (C) 1999 Landon Curt Noll
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##
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## Calc is open software; you can redistribute it and/or modify it under
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## the terms of the version 2.1 of the GNU Lesser General Public License
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## as published by the Free Software Foundation.
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##
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## Calc is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
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## Public License for more details.
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##
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## A copy of version 2.1 of the GNU Lesser General Public License is
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## distributed with calc under the filename COPYING-LGPL. You should have
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## received a copy with calc; if not, write to Free Software Foundation, Inc.
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## 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
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##
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## @(#) $Revision: 29.2 $
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## @(#) $Id: agd,v 29.2 2000/06/07 14:02:33 chongo Exp $
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## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/agd,v $
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##
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## Under source code control: 1997/09/06 20:03:34
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## File existed as early as: 1997
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##
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## chongo <was here> /\oo/\ http://www.isthe.com/chongo/
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## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
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