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<h3>
Raymond Toy pushed to branch rtoy-issue-78-unneded-code-code-in-complex-acos
at <a href="https://gitlab.common-lisp.net/cmucl/cmucl">cmucl / cmucl</a>
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Commits:
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<strong><a href="https://gitlab.common-lisp.net/cmucl/cmucl/-/commit/fe15b0a31119288135a67ab0a26f65b48e704fe8">fe15b0a3</a></strong>
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<span>by Raymond Toy</span>
<i>at 2020-05-24T08:10:39-07:00</i>
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<pre class="commit-message" style="white-space: pre-wrap; margin: 0;">Derive the values on the branch cuts for atanh
Conclusion: atanh is continuous with quadrant Iv for x > 1 and
quadrant II for x < -1. This, of course, is consistent with Kahan's
coutner-clowkwise continuity principla.
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<h4>1 changed file:</h4>
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<a href="#15fc200d41649e9bcd7e149a05640da3e3bc9f18">
src/code/irrat.lisp
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<h4>Changes:</h4>
<li id="15fc200d41649e9bcd7e149a05640da3e3bc9f18">
<a href="https://gitlab.common-lisp.net/cmucl/cmucl/-/commit/fe15b0a31119288135a67ab0a26f65b48e704fe8#15fc200d41649e9bcd7e149a05640da3e3bc9f18"><strong>src/code/irrat.lisp</strong></a>
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<td class="line_content match " style="padding-left: 0.5em; padding-right: 0.5em; color: rgba(0,0,0,0.3);" bgcolor="#fafafa">@@ -1449,7 +1449,44 @@ Z may be any number, but the result is always a complex."</td>
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<pre style="margin: 0;"> <span id="LC1449" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; +infinity, but the following code returns approx 176 + i*pi/4. The</span></span>
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<pre style="margin: 0;"> <span id="LC1450" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; reason for the imaginary part is caused by the fact that arg i*y is</span></span>
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<pre style="margin: 0;"> <span id="LC1451" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; never 0 since we have positive and negative zeroes.</span></span>
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<pre style="margin: 0;">-<span id="LC1452" class="line" lang="common_lisp"></span>
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<pre style="margin: 0;">+<span id="LC1452" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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<pre style="margin: 0;">+<span id="LC1453" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; The branch cut for atanh is on the real axis for x < -1 and x > 1.</span></span>
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<pre style="margin: 0;">+<span id="LC1454" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; Let's derive the values on the branch cut.</span></span>
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<pre style="margin: 0;">+<span id="LC1455" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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<pre style="margin: 0;">+<span id="LC1456" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; atanh(z) = 1/2*(log(1+z)-log(1-z))</span></span>
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<pre style="margin: 0;">+<span id="LC1457" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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<pre style="margin: 0;">+<span id="LC1458" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; For z = x, x > 1:</span></span>
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<pre style="margin: 0;">+<span id="LC1459" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; atanh(x) = 1/2*(log(1+x) - log(1-x))</span></span>
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<pre style="margin: 0;">+<span id="LC1460" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) - [log(x-1) + i*arg(1-x))</span></span>
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<pre style="margin: 0;">+<span id="LC1461" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) - (log(x-1) + i*pi))</span></span>
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<pre style="margin: 0;">+<span id="LC1462" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) - log(x-1) _ i*pi)</span></span>
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<pre style="margin: 0;">+<span id="LC1463" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*log((1+x)/(x-1)) - i*pi/2</span></span>
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<pre style="margin: 0;">+<span id="LC1464" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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<pre style="margin: 0;">+<span id="LC1465" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; For z = -x, x > 1</span></span>
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<pre style="margin: 0;">+<span id="LC1466" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; atanh(x) = 1/2*(log(1-x) - log(1+x))</span></span>
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<pre style="margin: 0;">+<span id="LC1467" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*((log(x-1) + i*arg(1-x)) - log(1+x))</span></span>
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<pre style="margin: 0;">+<span id="LC1468" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*((log(x-1) + i*pi) - log(1+x))</span></span>
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1469
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<pre style="margin: 0;">+<span id="LC1469" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log((x-1)/(x+1)) + i*pi)</span></span>
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1470
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<pre style="margin: 0;">+<span id="LC1470" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*log((x-1)/(x+1)) + i*pi/2</span></span>
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1471
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<pre style="margin: 0;">+<span id="LC1471" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1472
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<pre style="margin: 0;">+<span id="LC1472" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; For z = x - i0, x > 1</span></span>
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1473
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<pre style="margin: 0;">+<span id="LC1473" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; atanh(z) = 1/2*(log(1+x - i0) - log(1-x+i0))</span></span>
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1474
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<pre style="margin: 0;">+<span id="LC1474" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) + i*arg(1+x,-0) - (log(x-1) + i*arg(1-x, +0)))</span></span>
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1475
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<pre style="margin: 0;">+<span id="LC1475" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) - i*0 - (log(x-1) + i*pi))</span></span>
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1476
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<pre style="margin: 0;">+<span id="LC1476" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(1+x) - log(x-1) - i*pi)</span></span>
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1477
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<pre style="margin: 0;">+<span id="LC1477" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*log((1+x)/(x-1)) - i*pi/2</span></span>
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1478
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<pre style="margin: 0;">+<span id="LC1478" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1479
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<pre style="margin: 0;">+<span id="LC1479" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; This is the same answer we get for atanh(x), x > 1. Hence, atanh</span></span>
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1480
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<pre style="margin: 0;">+<span id="LC1480" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; is continuous with quadrant IV along the branch cut x > 1.</span></span>
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1481
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<pre style="margin: 0;">+<span id="LC1481" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1482
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<pre style="margin: 0;">+<span id="LC1482" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; Similary, for z = -x + i0, x > 1:</span></span>
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1483
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<pre style="margin: 0;">+<span id="LC1483" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; atanh(z) = 1/2*(log(1-x + i0) - log(1+x-i0))</span></span>
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1484
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<pre style="margin: 0;">+<span id="LC1484" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*((log(x-1) + i*arg(1-x, +0)) - (log(1+x)+i*arg(1+x, -0)))</span></span>
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1485
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<pre style="margin: 0;">+<span id="LC1485" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(x-1) + i*pi - (log(1+x) - i0))</span></span>
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1486
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<pre style="margin: 0;">+<span id="LC1486" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*(log(x-1) - log(1+x) + i*pi)</span></span>
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1487
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<pre style="margin: 0;">+<span id="LC1487" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 1/2*log((x-1)/(x+1)) + i*pi/2</span></span>
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1488
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<pre style="margin: 0;">+<span id="LC1488" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; This is same answer as atanh(x), x < -1. Thus, atanh is continuous</span></span>
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1489
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<pre style="margin: 0;">+<span id="LC1489" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; with quadrant II on the branch cut for x < -1</span></span>
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1453
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<pre style="margin: 0;"> <span id="LC1490" class="line" lang="common_lisp"><span class="p">(</span><span class="nb" style="color: #0086b3;">defun</span> <span class="nv" style="color: #008080;">complex-atanh</span> <span class="p">(</span><span class="nv" style="color: #008080;">z</span><span class="p">)</span></span>
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1454
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<pre style="margin: 0;"> <span id="LC1491" class="line" lang="common_lisp"> <span class="s" style="color: #d14;">"Compute atanh z = (log(1+z) - log(1-z))/2"</span></span>
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<pre style="margin: 0;"> <span id="LC1492" class="line" lang="common_lisp"> <span class="p">(</span><span class="k" style="font-weight: 600;">declare</span> <span class="p">(</span><span class="nc" style="color: #458; font-weight: 600;">number</span> <span class="nv" style="color: #008080;">z</span><span class="p">))</span></span>
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