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52 lines
1.3 KiB
Plaintext
52 lines
1.3 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 nonzero real, 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 nultiple 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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.86577 .8657694832 .865769483239659
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> print gd(2+1i, 1e-5), gd(2+1i, 1e-10)
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1.42291+.22751i 1.4229114625+.2275106584i
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LIMITS
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none
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LIBRARY
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COMPLEX *cgd(COMPLEX *x, NUMBER *eps)
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SEE ALSO
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agd, exp, ln, sin, sinh, etc.
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