133 lines
11 KiB
HTML
133 lines
11 KiB
HTML
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<title>std::riemann_zeta, std::riemann_zetaf, std::riemann_zetal</title>
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>riemann_zeta, <span style="font-size:0.7em; line-height:130%">std::</span>riemann_zetaf, <span style="font-size:0.7em; line-height:130%">std::</span>riemann_zetal</h1>
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<div id="bodyContent">
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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-special_functions.html">special functions</a></span></div>
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<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
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<div>定义于头文件 <code><a href="cpp-header-cmath.html"><cmath></a></code></div>
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<td></td>
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<td></td>
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<tr class="t-dcl t-since-cxx17">
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> riemann_zeta<span class="br0">(</span> <span class="kw4">double</span> arg <span class="br0">)</span><span class="sy4">;</span><br /></span>
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<p><span class="kw4">float</span> riemann_zeta<span class="br0">(</span> <span class="kw4">float</span> arg <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">long</span> <span class="kw4">double</span> riemann_zeta<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> arg <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">float</span> riemann_zetaf<span class="br0">(</span> <span class="kw4">float</span> arg <span class="br0">)</span><span class="sy4">;</span><br /></p>
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<span class="kw4">long</span> <span class="kw4">double</span> riemann_zetal<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> arg <span class="br0">)</span><span class="sy4">;</span></div>
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<td>(1)</td>
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<td><span class="t-mark-rev t-since-cxx17">(C++17 起)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> riemann_zeta<span class="br0">(</span> IntegralType arg <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(2)</td>
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<td><span class="t-mark-rev t-since-cxx17">(C++17 起)</span></td>
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<div class="t-li1"><span class="t-li">1)</span> 计算 <code>arg</code> 的<a href="https://en.wikipedia.org/wiki/Riemann_zeta_function" class="extiw" title="enwiki:Riemann zeta function">黎曼 zeta 函数</a>。</div>
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<div class="t-li1"><span class="t-li">2)</span> 接受任何<a href="cpp-types-is_integral.html">整数类型</a>实参的重载集或函数模板。等价于 <span class="t-v">(1)</span> ，在将参数转型为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 后。</div>
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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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<td>arg</td>
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<td>-</td>
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<td>浮点或整数类型的值</td>
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</tr>
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</table>
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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>若无错误发生，则为 <code>arg</code> 的黎曼 Zeta 函数值， <span class="texhtml" style="white-space: nowrap;">ζ(arg)</span> ，定义于整个实轴：</p>
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<ul>
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<li>对于 <span class="texhtml" style="white-space: nowrap;">arg>1</span> ，为 <span class="texhtml" style="white-space: nowrap;">Σ<span class="t-su t-su-b">∞<br />
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n=1</span>n<span class="t-su">-arg<br /></span></span></li>
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<li>对于 <span class="texhtml" style="white-space: nowrap;">0≤arg≤1</span> ，为 <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<td>1</td>
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</tr>
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<tr>
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<td>1-2<span class="t-su">1-arg<br /></span></td>
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</tr>
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Σ<span class="t-su t-su-b">∞<br />
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n=1</span>(-1)<span class="t-su">n-1<br /></span>n<span class="t-su">-arg<br /></span></li>
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<li>对于 <span class="texhtml" style="white-space: nowrap;">arg<0</span> ，为 <span class="texhtml" style="white-space: nowrap;">2<span class="t-su">arg<br /></span>π<span class="t-su">arg-1<br /></span>sin(<span class="t-mfrac"></span></span>
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<table>
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<td>πarg</td>
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<td>2</td>
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</table>
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)Γ(1−arg)ζ(1−arg)</li>
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</ul>
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">错误处理</span></h3>
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<p>可能报告 <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> 中指定的错误</p>
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<ul>
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<li>若参数是 NaN ，则返回 NaN 且不报告定义域错误</li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">注解</span></h3>
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<p>不支持 C++17 ，但支持 <a href="cpp-experimental-special_functions.html">ISO 29124:2010</a> 的实现会提供此函数，若实现定义了 <code>__STDCPP_MATH_SPEC_FUNCS__</code> 为至少 201003L 的值，且用户在包含任何标准库头文件前定义了 <code>__STDCPP_WANT_MATH_SPEC_FUNCS__</code> 。</p>
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<p>不支持 ISO 29124:2010 但支持 TR 19768:2007 (TR1) 的实现，在头文件 <code>tr1/cmath</code> 及命名空间 <code>std::tr1</code> 中提供此函数。</p>
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<p>此函数的一种实现<a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/math_toolkit/zetas/zeta.html">亦可用于 boost.math</a></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="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 <cmath></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="co1">// 广为人知的值的点检查</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">"ζ(-1) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span><span class="sy2">-</span><span class="nu0">1</span><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">"ζ(0) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span><span class="nu0">0</span><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">"ζ(1) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span><span class="nu0">1</span><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">"ζ(0.5) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span><span class="nu16">0.5</span><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">"ζ(2) = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> <span class="st0">"(π²/6 = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-math-pow.html"><span class="kw1332">std::<span class="me2">pow</span></span></a><span class="br0">(</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="nu0">2</span><span class="br0">)</span><span class="sy2">/</span><span class="nu0">6</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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</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">ζ(-1) = -0.0833333
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ζ(0) = -0.5
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ζ(1) = inf
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ζ(0.5) = -1.46035
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ζ(2) = 1.64493 (π²/6 = 1.64493)</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">外部链接</span></h3>
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<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RiemannZetaFunction.html">Weisstein, Eric W. “黎曼 Zeta 函数”</a>来自 MathWorld--A Wolfram Web Resource 。</p>
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</div>
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<div class="visualClear"></div>
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