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@@ -1726,6 +1726,46 @@ Z may be any number, but the result is always a complex." |
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1726
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1726
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(asinh (imagpart (* (conjugate sqrt-1+z)
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1727
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1727
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sqrt-1-z))))))))
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1728
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1728
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1729
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+;; acosh(z) = 2*log(sqrt((x+1)/2) + sqrt((x-1)/2))
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1730
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+;;
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1731
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+;; For z = x, 0 <= x < 1
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1732
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+;; acosh(z) = 2*log(sqrt((x+1)/2) + sqrt((x-1)/2))
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1733
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+;; = 2*log(sqrt((x+1)/2) + i*sqrt((1-x)/2))
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1734
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+;; = 2*(log(1) + i*arg(sqrt((x+1)/2) + i*sqrt((1-x)/2)))
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1735
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+;; = 2*i*atan(sqrt((1-x)/2), sqrt((x+1)/2))
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1736
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+;; = 2*i*atan(sqrt((1-x)/(1+x)))
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1737
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+;;
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1738
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+;; For z = -x, x > 1
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1739
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+;; acosh(z) = 2*log(sqrt((1-x)/2) + sqrt((-x-1)/2))
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1740
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+;; = 2*log((i*sqrt((x-1)/2) + 0 + i*sqrt((1+x)/2)) + 0)
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1741
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+;; = 2*log(i*(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)
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1742
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+;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*arg(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)
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1743
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+;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi/2)
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1744
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+;; = 2*log(x+sqrt(x+1)*sqrt(x-1)) + i*pi
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1745
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+;;
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1746
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+;; For z = x + i0, 0 <= x < 1
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1747
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+;; acosh(z) = 2*log(sqrt((1+x)/2+i0) + sqrt((x-1)/2+i0))
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1748
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+;; = 2*log(sqrt((1+x)/2)+i0 + i*sqrt((1-x)/2) + 0)
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1749
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+;; = 2*log(sqrt((1+x)/2) + i*sqrt((1-x)/2))
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1750
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+;; = 2*(log(1) + i*arg(sqrt((1+x)/2) + i*sqrt((1-x)/2))
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1751
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+;; = 0 + 2*i*atan(sqrt((1-x)/2)/sqrt((1+x)/2)) +
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1752
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+;; = 0 + 2*i*atan(sqrt((1-x)/(1+x))
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1753
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+;;
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1754
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+;; This is the same value we got for acosh(x), 0 <= x < 1. Hence,
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1755
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+;; acosh is continuous with quadrant I on the branch cut for 0 <= x <
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1756
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+;; 1.
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1757
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+;;
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1758
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+;; Finally, for z = -x + i0, x > 1
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1759
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+;; acosh(z) = 2*log(sqrt((1-x)/2+i0) + sqrt((-x-1)/2+i0))
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1760
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+;; = 2*log(i*sqrt((x-1)/2) + 0 + i*sqrt((1+x)/2 + i0))
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1761
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+;; = 2*log(i*(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)
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1762
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+;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*arg(0, (sqrt((x-1)/2) + sqrt((1+x)/2)))
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1763
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+;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi/2)
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1764
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+;; = 2*log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi
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1765
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+;;
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1766
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+;; We see that this is the same expression for acosh(z), z < -1.
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1767
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+;; Hence, acosh(z) is continuous with quadrant II on the branch cut x
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1768
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+;; < -1.
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1729
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1769
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(defun complex-acosh (z)
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1730
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1770
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"Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))
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1731
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1771
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