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58 lines
1.4 KiB
Plaintext
58 lines
1.4 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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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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