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1c839dfede
Add new logn(x, n [,eps]) builtin to compute logarithms to base n. Verify that eps arguments (error tolerance arguments that override the default epsilon value) to builtin functions have proper values. Previously the eps argument had little to no value checks for many builtin functions. Document in help files for builtin functions that take eps arguments, the LIMIT range for such eps values.
74 lines
2.3 KiB
Plaintext
74 lines
2.3 KiB
Plaintext
NAME
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gd - Gudermannian function
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SYNOPSIS
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gd(z [,eps])
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TYPES
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z number (real or complex)
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eps 0 < real < 1, defaults to epsilon()
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return number or "Log of zero or infinity" error value
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DESCRIPTION
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Calculate the Gudermannian of z to a multiple of eps with errors in
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real and imaginary parts less in absolute value than .75 * eps,
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or return an error value if z is close to one of the branch points
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at odd multiples of (pi/2) * i.
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gd(z) is usually defined initially for real z by one of the formulae
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gd(z) = 2 * atan(exp(z)) - pi/2
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= 2 * atan(tanh(z/2))
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= atan(sinh(z)),
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or as the integral from 0 to z of (1/cosh(t))dt. For complex z, the
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principal branch, approximated by gd(z, eps), has the cut:
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re(z) = 0, abs(im(z)) >= pi/2; on the cut calc takes gd(z) to be
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the limit as z is approached from the right or left according as
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im(z) > or < 0.
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If z = x + y*i and abs(y) < pi/2, gd(z) is given by
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gd(z) = atan(sinh(x)/cos(y)) + i * atanh(sin(y)/cosh(x)).
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EXAMPLE
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; print gd(1, 1e-5), gd(1, 1e-10), gd(1, 1e-15)
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0.86577 0.8657694832 0.865769483239659
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; print gd(2+1i, 1e-5), gd(2+1i, 1e-10)
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1.42291+0.22751i 1.4229114625+0.2275106584i
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LIMITS
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0 < eps < 1
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LINK LIBRARY
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COMPLEX *c_gd(COMPLEX *x, NUMBER *eps)
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SEE ALSO
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agd, exp, ln, sin, sinh, etc.
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## Copyright (C) 1999,2021,2023 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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## 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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##
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## Under source code control: 1997/09/06 20:03:35
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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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