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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/00d0cb6f5f931b01000bef17626d5604216de8b0">00d0cb6f</a></strong>
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<span>by Raymond Toy</span>
<i>at 2020-05-24T10:17:44-07:00</i>
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<pre class="commit-message" style="white-space: pre-wrap; margin: 0;">Derive the values on the branch cuts for acosh
Conclusion: on the branch cuts, acosh is continuous with quadrant I
for 9 <= x < 1 and with 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/00d0cb6f5f931b01000bef17626d5604216de8b0#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">@@ -1726,6 +1726,46 @@ Z may be any number, but the result is always a complex."</td>
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<pre style="margin: 0;"> <span id="LC1726" class="line" lang="common_lisp"> <span class="p">(</span><span class="nb" style="color: #0086b3;">asinh</span> <span class="p">(</span><span class="nb" style="color: #0086b3;">imagpart</span> <span class="p">(</span><span class="nb" style="color: #0086b3;">*</span> <span class="p">(</span><span class="nb" style="color: #0086b3;">conjugate</span> <span class="nv" style="color: #008080;">sqrt-1+z</span><span class="p">)</span></span>
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<pre style="margin: 0;"> <span id="LC1727" class="line" lang="common_lisp"> <span class="nv" style="color: #008080;">sqrt-1-z</span><span class="p">))))))))</span></span>
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<pre style="margin: 0;"> <span id="LC1728" class="line" lang="common_lisp"></span>
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<pre style="margin: 0;">+<span id="LC1729" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh(z) = 2*log(sqrt((x+1)/2) + sqrt((x-1)/2))</span></span>
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<pre style="margin: 0;">+<span id="LC1730" 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="LC1731" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; For z = x, 0 <= x < 1</span></span>
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<pre style="margin: 0;">+<span id="LC1732" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh(z) = 2*log(sqrt((x+1)/2) + sqrt((x-1)/2))</span></span>
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<pre style="margin: 0;">+<span id="LC1733" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(sqrt((x+1)/2) + i*sqrt((1-x)/2))</span></span>
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<pre style="margin: 0;">+<span id="LC1734" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(1) + i*arg(sqrt((x+1)/2) + i*sqrt((1-x)/2)))</span></span>
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<pre style="margin: 0;">+<span id="LC1735" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*i*atan(sqrt((1-x)/2), sqrt((x+1)/2))</span></span>
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<pre style="margin: 0;">+<span id="LC1736" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*i*atan(sqrt((1-x)/(1+x)))</span></span>
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<pre style="margin: 0;">+<span id="LC1737" 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="LC1738" 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="LC1739" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh(z) = 2*log(sqrt((1-x)/2) + sqrt((-x-1)/2))</span></span>
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<pre style="margin: 0;">+<span id="LC1740" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log((i*sqrt((x-1)/2) + 0 + i*sqrt((1+x)/2)) + 0)</span></span>
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<pre style="margin: 0;">+<span id="LC1741" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(i*(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)</span></span>
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<pre style="margin: 0;">+<span id="LC1742" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*arg(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)</span></span>
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<pre style="margin: 0;">+<span id="LC1743" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi/2)</span></span>
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<pre style="margin: 0;">+<span id="LC1744" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(x+sqrt(x+1)*sqrt(x-1)) + i*pi</span></span>
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<pre style="margin: 0;">+<span id="LC1745" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1746
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<pre style="margin: 0;">+<span id="LC1746" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; For z = x + i0, 0 <= x < 1</span></span>
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1747
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<pre style="margin: 0;">+<span id="LC1747" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh(z) = 2*log(sqrt((1+x)/2+i0) + sqrt((x-1)/2+i0))</span></span>
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1748
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<pre style="margin: 0;">+<span id="LC1748" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(sqrt((1+x)/2)+i0 + i*sqrt((1-x)/2) + 0)</span></span>
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1749
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<pre style="margin: 0;">+<span id="LC1749" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(sqrt((1+x)/2) + i*sqrt((1-x)/2))</span></span>
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1750
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<pre style="margin: 0;">+<span id="LC1750" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(1) + i*arg(sqrt((1+x)/2) + i*sqrt((1-x)/2))</span></span>
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1751
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<pre style="margin: 0;">+<span id="LC1751" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 0 + 2*i*atan(sqrt((1-x)/2)/sqrt((1+x)/2)) + </span></span>
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1752
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<pre style="margin: 0;">+<span id="LC1752" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 0 + 2*i*atan(sqrt((1-x)/(1+x))</span></span>
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1753
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<pre style="margin: 0;">+<span id="LC1753" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1754
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<pre style="margin: 0;">+<span id="LC1754" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; This is the same value we got for acosh(x), 0 <= x < 1. Hence,</span></span>
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1755
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<pre style="margin: 0;">+<span id="LC1755" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh is continuous with quadrant I on the branch cut for 0 <= x <</span></span>
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1756
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<pre style="margin: 0;">+<span id="LC1756" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; 1.</span></span>
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1757
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<pre style="margin: 0;">+<span id="LC1757" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1758
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<pre style="margin: 0;">+<span id="LC1758" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; Finally, for z = -x + i0, x > 1</span></span>
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1759
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<pre style="margin: 0;">+<span id="LC1759" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; acosh(z) = 2*log(sqrt((1-x)/2+i0) + sqrt((-x-1)/2+i0))</span></span>
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1760
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<pre style="margin: 0;">+<span id="LC1760" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(i*sqrt((x-1)/2) + 0 + i*sqrt((1+x)/2 + i0))</span></span>
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1761
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<pre style="margin: 0;">+<span id="LC1761" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(i*(sqrt((x-1)/2) + sqrt((1+x)/2)) + 0)</span></span>
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1762
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<pre style="margin: 0;">+<span id="LC1762" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*arg(0, (sqrt((x-1)/2) + sqrt((1+x)/2)))</span></span>
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1763
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<pre style="margin: 0;">+<span id="LC1763" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*(log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi/2)</span></span>
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1764
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<pre style="margin: 0;">+<span id="LC1764" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; = 2*log(sqrt((x-1)/2) + sqrt((1+x)/2)) + i*pi</span></span>
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1765
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<pre style="margin: 0;">+<span id="LC1765" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;;</span></span>
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1766
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<pre style="margin: 0;">+<span id="LC1766" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; We see that this is the same expression for acosh(z), z < -1.</span></span>
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1767
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<pre style="margin: 0;">+<span id="LC1767" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; Hence, acosh(z) is continuous with quadrant II on the branch cut x</span></span>
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1768
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<pre style="margin: 0;">+<span id="LC1768" class="line" lang="common_lisp"><span class="c1" style="color: #998; font-style: italic;">;; < -1.</span></span>
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<pre style="margin: 0;"> <span id="LC1769" class="line" lang="common_lisp"><span class="p">(</span><span class="nb" style="color: #0086b3;">defun</span> <span class="nv" style="color: #008080;">complex-acosh</span> <span class="p">(</span><span class="nv" style="color: #008080;">z</span><span class="p">)</span></span>
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<pre style="margin: 0;"> <span id="LC1770" class="line" lang="common_lisp"> <span class="s" style="color: #d14;">"Compute acosh z = 2 * log(sqrt((z+1)/2) + sqrt((z-1)/2))</span></span>
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1771
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<pre style="margin: 0;"> <span id="LC1771" class="line" lang="common_lisp"></span>
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