134 lines
11 KiB
HTML
134 lines
11 KiB
HTML
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<title>std::exp(std::complex)</title>
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>exp<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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<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> exp<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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<p>计算 <code>z</code> 的底 e 指数，即 <i>e</i> （欧拉数， <code>2.7182818</code> ）的 <code>z</code> 次幂。</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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<td>z</td>
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<td>-</td>
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<td>复数值</td>
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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>若不出现错误，则返回 <i>e</i> 的 <code>z</code> 次幂， <span class="texhtml" style="white-space: nowrap;">e<span class="t-su">z<br /></span></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-exp.html"><span class="kw1321">std::<span class="me2">exp</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-exp.html"><span class="kw1321">std::<span class="me2">exp</span></span></a><span class="br0">(</span>z<span class="br0">)</span><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>(1,+0)</code> 。</li>
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<li>若 <code>z</code> 为 <code>(x,+∞)</code> （对于任何有限 x ），则结果为 <code>(NaN,NaN)</code> 并引发 <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>(x,NaN)</code> （对于任何有限 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>(+∞,+0)</code> ，则结果为 <code>(+∞,+0)</code> 。</li>
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<li>若 <code>z</code> 为 <code>(-∞,y)</code> （对于任何有限 y ），则结果为 <code>+0cis(y)</code> 。</li>
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<li>若 <code>z</code> 为 <code>(+∞,y)</code> （对于任何有限非零 y ），则结果为 <code>+∞cis(y)</code> 。</li>
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<li>若 <code>z</code> 为 <code>(-∞,+∞)</code> ，则结果为 <code>(±0,±0)</code> （符号未指定）。</li>
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<li>若 <code>z</code> 为 <code>(+∞,+∞)</code> ，则结果为 <code>(±∞,NaN)</code> 并引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span> （实部符号未指定）</li>
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<li>若 <code>z</code> 为 <code>(-∞,NaN)</code> ，则结果为 <code>(±0,±0)</code> （符号未指定）</li>
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<li>若 <code>z</code> 为 <code>(+∞,NaN)</code> ，则结果为 <code>(±∞,NaN)</code> （实部符号未指定）</li>
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<li>若 <code>z</code> 为 <code>(NaN,+0)</code> ，则结果为 <code>(NaN,+0)</code></li>
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<li>若 <code>z</code> 为 <code>(NaN,y)</code> （对于任何非零 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,NaN)</code> ，则结果为 <code>(NaN,NaN)</code> 。</li>
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</ul>
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<p>其中 <span class="texhtml" style="white-space: nowrap;">cis(y)</span> 为 <span class="texhtml" style="white-space: nowrap;">cos(y) + i sin(y)</span> 。</p>
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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>复指数函数 <span class="texhtml" style="white-space: nowrap;">e<span class="t-su">z<br /></span></span> 对于自变量 <span class="texhtml" style="white-space: nowrap;">z = x+iy</span> 的值等于 <span class="texhtml" style="white-space: nowrap;">e<span class="t-su">x<br /></span> cis(y)</span> ，或 <span class="texhtml" style="white-space: nowrap;">e<span class="t-su">x<br /></span> (cos(y) + i sin(y))</span> 。</p>
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<p>指数函数在复平面上是<i>整函数</i>且无分支切割。</p>
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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="cpp source-cpp">
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<pre class="de1"><span class="co2">#include <complex></span>
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<span class="co2">#include <iostream></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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<span class="kw4">const</span> <span class="kw4">double</span> pi <span class="sy1">=</span> <a href="cpp-numeric-math-acos.html"><span class="kw1337">std::<span class="me2">acos</span></span></a><span class="br0">(</span><span class="sy2">-</span><span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw4">const</span> <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> <a href="cpp-io-manip-fixed.html"><span class="kw1784">std::<span class="me2">fixed</span></span></a> <span class="sy1"><<</span> <span class="st0">" exp(i*pi) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-exp.html"><span class="kw1321">std::<span class="me2">exp</span></span></a><span class="br0">(</span>i <span class="sy2">*</span> pi<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="br0">}</span></pre></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">exp(i*pi) = (-1.000000,0.000000)</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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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-complex-log.html"><span class="t-lines"><span>log<span class="t-dsc-small">(std::complex)</span></span></span></a></div>
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</div>
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<td>沿负实轴切割的复自然对数<br />
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<span class="t-mark">(函数模板)</span></td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-math-exp.html"><span class="t-lines"><span>exp</span><span>expf</span><span>expl</span></span></a></div>
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<div><span class="t-lines"><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>返回 <i>e</i> 的给定次幂（ <span class="mjax" style="display:none">\({\small e^x}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">e<sup>x</sup></span> ）<br />
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<span class="t-mark">(函数)</span></td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-valarray-exp.html"><span class="t-lines"><span>exp<span class="t-dsc-small">(std::valarray)</span></span></span></a></div>
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</div>
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<td>应用函数 <span class="t-lc"><a href="cpp-numeric-math-exp.html">std::exp</a></span> 到 valarray 的每个元素<br />
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<span class="t-mark">(函数模板)</span></td>
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