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@@ -1449,7 +1449,44 @@ Z may be any number, but the result is always a complex." |
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1449
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1449
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;; +infinity, but the following code returns approx 176 + i*pi/4. The
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1450
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1450
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;; reason for the imaginary part is caused by the fact that arg i*y is
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1451
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1451
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;; never 0 since we have positive and negative zeroes.
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1452
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-
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1452
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+;;
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1453
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+;; The branch cut for atanh is on the real axis for x < -1 and x > 1.
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1454
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+;; Let's derive the values on the branch cut.
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1455
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+;;
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1456
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+;; atanh(z) = 1/2*(log(1+z)-log(1-z))
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1457
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+;;
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1458
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+;; For z = x, x > 1:
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1459
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+;; atanh(x) = 1/2*(log(1+x) - log(1-x))
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1460
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+;; = 1/2*(log(1+x) - [log(x-1) + i*arg(1-x))
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1461
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+;; = 1/2*(log(1+x) - (log(x-1) + i*pi))
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1462
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+;; = 1/2*(log(1+x) - log(x-1) _ i*pi)
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1463
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+;; = 1/2*log((1+x)/(x-1)) - i*pi/2
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1464
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+;;
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1465
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+;; For z = -x, x > 1
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1466
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+;; atanh(x) = 1/2*(log(1-x) - log(1+x))
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1467
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+;; = 1/2*((log(x-1) + i*arg(1-x)) - log(1+x))
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1468
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+;; = 1/2*((log(x-1) + i*pi) - log(1+x))
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1469
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+;; = 1/2*(log((x-1)/(x+1)) + i*pi)
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1470
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+;; = 1/2*log((x-1)/(x+1)) + i*pi/2
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1471
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+;;
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1472
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+;; For z = x - i0, x > 1
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1473
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+;; atanh(z) = 1/2*(log(1+x - i0) - log(1-x+i0))
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1474
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+;; = 1/2*(log(1+x) + i*arg(1+x,-0) - (log(x-1) + i*arg(1-x, +0)))
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1475
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+;; = 1/2*(log(1+x) - i*0 - (log(x-1) + i*pi))
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1476
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+;; = 1/2*(log(1+x) - log(x-1) - i*pi)
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1477
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+;; = 1/2*log((1+x)/(x-1)) - i*pi/2
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1478
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+;;
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1479
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+;; This is the same answer we get for atanh(x), x > 1. Hence, atanh
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1480
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+;; is continuous with quadrant IV along the branch cut x > 1.
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1481
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+;;
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1482
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+;; Similary, for z = -x + i0, x > 1:
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1483
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+;; atanh(z) = 1/2*(log(1-x + i0) - log(1+x-i0))
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1484
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+;; = 1/2*((log(x-1) + i*arg(1-x, +0)) - (log(1+x)+i*arg(1+x, -0)))
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1485
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+;; = 1/2*(log(x-1) + i*pi - (log(1+x) - i0))
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1486
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+;; = 1/2*(log(x-1) - log(1+x) + i*pi)
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1487
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+;; = 1/2*log((x-1)/(x+1)) + i*pi/2
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1488
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+;; This is same answer as atanh(x), x < -1. Thus, atanh is continuous
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1489
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+;; with quadrant II on the branch cut for x < -1
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1453
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1490
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(defun complex-atanh (z)
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1454
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1491
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"Compute atanh z = (log(1+z) - log(1-z))/2"
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1455
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1492
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(declare (number z))
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