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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>atanh<span style="font-size:0.7em; line-height:130%">(std::complex)</span></h1>
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<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a>‎ | <a href="cpp-numeric-complex.html">complex</a></span></div>
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<div>定义于头文件 <code><a href="cpp-header-complex.html"><complex></a></code></div>
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<td></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw1">template</span><span class="sy1"><</span> <span class="kw1">class</span> T <span class="sy1">></span><br />
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complex<span class="sy1"><</span>T<span class="sy1">></span> atanh<span class="br0">(</span> <span class="kw4">const</span> complex<span class="sy1"><</span>T<span class="sy1">></span><span class="sy3">&</span> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td class="t-dcl-nopad"></td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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<p>计算 <code>z</code> 的复反双曲正切，其分支切割为沿实轴的 <span class="texhtml" style="white-space: nowrap;">[−1; +1]</span> 区间外部。</p>
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<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
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<table class="t-par-begin">
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<tr class="t-par">
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<td>z</td>
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<td>-</td>
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<td>复数值</td>
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</tr>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
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<p>若不发生错误，则返回 <code>z</code> 的复反双曲正切，范围在数学上为沿着实轴无界的半条，沿着虚轴为区间 <span class="texhtml" style="white-space: nowrap;">[−iπ/2; +iπ/2]</span> 。</p>
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86.E5.8F.8A.E7.89.B9.E6.AE.8A.E5.80.BC">错误处理及特殊值</span></h3>
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<p>报告的错误与 <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> 一致。</p>
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<p>若实现支持 IEEE 浮点算术，则</p>
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<ul>
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<li><span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>std<span class="sy4">::</span><span class="me2">conj</span><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span> <span class="sy1">==</span> std<span class="sy4">::</span><span class="me2">conj</span><span class="br0">(</span><a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>z<span class="br0">)</span><span class="br0">)</span></span></span></li>
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<li><span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span><span class="sy2">-</span>z<span class="br0">)</span> <span class="sy1">==</span> <span class="sy2">-</span><a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>z<span class="br0">)</span></span></span></li>
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<li>若 <code>z</code> 为 <code>(+0,+0)</code> ，则结果为 <code>(+0,+0)</code></li>
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<li>若 <code>z</code> 为 <code>(+0,NaN)</code> ，则结果为 <code>(+0,NaN)</code></li>
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<li>若 <code>z</code> 为 <code>(+1,+0)</code> ，则结果为 <code>(+∞,+0)</code> 并引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_DIVBYZERO</a></span></li>
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<li>若 <code>z</code> 为 <code>(x,+∞)</code> （对于任何有限正 x ），则结果为 <code>(+0,π/2)</code></li>
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<li>若 <code>z</code> 为 <code>(x,NaN)</code> （对于任何有限非零 x ），则结果为 <code>(NaN,NaN)</code> 并可能引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 <code>z</code> 为 <code>(+∞,y)</code> （对于任何有限正 y ），结果为 <code>(+0,π/2)</code></li>
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<li>若 <code>z</code> 为 <code>(+∞,+∞)</code> ，则结果为 <code>(+0,π/2)</code></li>
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<li>若 <code>z</code> 为 <code>(+∞,NaN)</code> ，则结果为 <code>(+0,NaN)</code></li>
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<li>若 <code>z</code> 为 <code>(NaN,y)</code> （对于任何有限的 y ） ，则结果为 <code>(NaN,NaN)</code> 并可能引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 <code>z</code> 为 <code>(NaN,+∞)</code> ，则结果为 <code>(±0,π/2)</code> （实部的符号未指定）</li>
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<li>若 <code>z</code> 为 <code>(NaN,NaN)</code> ，则结果为 <code>(NaN,NaN)</code></li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E6.84.8F">注意</span></h3>
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<p>尽管 C++ 标准命名此函数为“复弧双曲正切”，双曲函数的反函数却是面积函数。其参数是双曲扇形的面积，而非弧长。正确的名称是“复反双曲正切”，和较少见的“复面积双曲正切”。</p>
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<p>反双曲正切是多值函数，并要求复平面上的分支切割。我们约定将分支切割置于实轴的划分线 <span class="texhtml" style="white-space: nowrap;">(-∞,-1]</span> 和 <span class="texhtml" style="white-space: nowrap;">[+1,+∞)</span> 。</p>
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反双曲正切的主值的数学定义是 <span class="texhtml" style="white-space: nowrap;">atanh z = <span class="t-mfrac"></span></span>
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<table>
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<td>ln(1+z)-ln(z-1)</td>
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<td>2</td>
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。
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<p><br /></p>
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对于任何 z ， <span class="texhtml" style="white-space: nowrap;">atanh(z) = <span class="t-mfrac"></span></span>
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<td>atan(iz)</td>
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<td>i</td>
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</table>
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。
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="cpp source-cpp">
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<pre class="de1"><span class="co2">#include <iostream></span>
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<span class="co2">#include <complex></span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
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<span class="br0">{</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1784">std::<span class="me2">fixed</span></span></a><span class="sy4">;</span>
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<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z1<span class="br0">(</span><span class="nu0">2</span>, <span class="nu0">0</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"atanh"</span> <span class="sy1"><<</span> z1 <span class="sy1"><<</span> <span class="st0">" = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>z1<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z2<span class="br0">(</span><span class="nu0">2</span>, <span class="sy2">-</span><span class="nu16">0.0</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"atanh"</span> <span class="sy1"><<</span> z2 <span class="sy1"><<</span> <span class="st0">" (the other side of the cut) = "</span>
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<span class="sy1"><<</span> <a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>z2<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="co1">// 对于任何 z ， atanh(z) = atanh(iz)/i</span>
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<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> z3<span class="br0">(</span><span class="nu0">1</span>,<span class="nu0">2</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-numeric-complex.html"><span class="kw1472">std::<span class="me2">complex</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> i<span class="br0">(</span><span class="nu0">0</span>,<span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"atanh"</span> <span class="sy1"><<</span> z3 <span class="sy1"><<</span> <span class="st0">" = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-atanh.html"><span class="kw1345">std::<span class="me2">atanh</span></span></a><span class="br0">(</span>z3<span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
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<span class="sy1"><<</span> <span class="st0">"atan"</span> <span class="sy1"><<</span> z3<span class="sy2">*</span>i <span class="sy1"><<</span> <span class="st0">"/i = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-atan.html"><span class="kw1338">std::<span class="me2">atan</span></span></a><span class="br0">(</span>z3<span class="sy2">*</span>i<span class="br0">)</span><span class="sy2">/</span>i <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>输出：</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">atanh(2.000000,0.000000) = (0.549306,1.570796)
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atanh(2.000000,-0.000000) (the other side of the cut) = (0.549306,-1.570796)
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atanh(1.000000,2.000000) = (0.173287,1.178097)
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atan(-2.000000,1.000000)/i = (0.173287,1.178097)</pre></div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-complex-asinh.html"><span class="t-lines"><span>asinh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>计算复数的反双曲正弦（ <span class="mjax" style="display:none">\({\small\operatorname{arsinh}{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">arsinh(z)</span> ）<br />
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<span class="t-mark">(函数模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-complex-acosh.html"><span class="t-lines"><span>acosh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>计算复数的反双曲余弦（ <span class="mjax" style="display:none">\({\small\operatorname{arcosh}{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">arcosh(z)</span> ）<br />
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<span class="t-mark">(函数模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-complex-tanh.html"><span class="t-lines"><span>tanh<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
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</div>
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</td>
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<td>计算复数的双曲正切（ <span class="mjax" style="display:none">\({\small\tanh{z} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">tanh(z)</span> ）<br />
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<span class="t-mark">(函数模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-math-atanh.html"><span class="t-lines"><span>atanh</span><span>atanhf</span><span>atanhl</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>计算反双曲正切（ <span class="mjax" style="display:none">\({\small\operatorname{artanh}{x} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">artanh(x)</span> ）<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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</table>
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</div>
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<div class="visualClear"></div>
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</div>
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</div>
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</div>
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</body>
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</html>
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