145 lines
13 KiB
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145 lines
13 KiB
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<title>std::sph_legendre, std::sph_legendref, std::sph_legendrel</title>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_special_functions_sph_legendre skin-cppreference2 action-view cpp-navbar">
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>sph_legendre, <span style="font-size:0.7em; line-height:130%">std::</span>sph_legendref, <span style="font-size:0.7em; line-height:130%">std::</span>sph_legendrel</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-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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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> sph_legendre <span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, <span class="kw4">double</span> t <span class="br0">)</span><span class="sy4">;</span><br /></span>
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<p><span class="kw4">float</span> sph_legendre <span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, <span class="kw4">float</span> t <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">long</span> <span class="kw4">double</span> sph_legendre <span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, <span class="kw4">long</span> <span class="kw4">double</span> t <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">float</span> sph_legendref<span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, <span class="kw4">float</span> t <span class="br0">)</span><span class="sy4">;</span><br /></p>
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<span class="kw4">long</span> <span class="kw4">double</span> sph_legendrel<span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, <span class="kw4">long</span> <span class="kw4">double</span> t <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> sph_legendre <span class="br0">(</span> <span class="kw4">unsigned</span> l, <span class="kw4">unsigned</span> m, IntegralType t <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>l</code> 次、 <code>m</code> 阶和极角 <code>t</code> 的球关联勒让德函数。</div>
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<div class="t-li1"><span class="t-li">2)</span> 接受任意<a href="cpp-types-is_integral.html">整数类型</a>参数的重载集或函数模板。等价于将参数转型到 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 后的 <span class="t-v">(1)</span> 。</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>l</td>
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<td>-</td>
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<td>次数</td>
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</tr>
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<tr class="t-par">
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<td>m</td>
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<td>-</td>
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<td>阶数</td>
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</tr>
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<tr class="t-par">
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<td>t</td>
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<td>-</td>
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<td>极角，以弧度度量</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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若无错误发生，则返回 <code>l</code> 、 <code>m</code> 和 <code>t</code> 的球关联勒让德函数（即 ϕ = 0 下的球谐函数）的值，其中球谐函数定义为 <span class="texhtml" style="white-space: nowrap;">Y<span class="t-su t-su-b">m<br />
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l</span>(t,ϕ) = (-1)<span class="t-su">m<br /></span>[<span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>(2l+1)(l-m)!</td>
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</tr>
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<tr>
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<td>4π(l+m)!</td>
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</tr>
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</table>
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]<span class="t-su">1/2<br /></span>P<span class="t-su t-su-b">m<br />
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l</span>(cost)e<span class="t-su">imϕ<br /></span> ，其中 <span class="texhtml" style="white-space: nowrap;">P<span class="t-su t-su-b">m<br />
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l</span>(x)</span> 为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-special_math-assoc_legendre.html"><span class="kw1390">std::<span class="me2">assoc_legendre</span></span></a><span class="br0">(</span>l,m,x<span class="br0">)</span></span></span>) 且 <span class="texhtml" style="white-space: nowrap;">|m|≤l</span>
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<p>注意此函数定义包含 <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Condon-ShortleyPhase.html">Condon-Shortley 相位项</a> <span class="texhtml" style="white-space: nowrap;">(-1)<span class="t-su">m<br /></span></span> ，因为以 <span class="t-lc"><a href="cpp-numeric-special_math-assoc_legendre.html" class="mw-redirect">std::assoc_legendre</a></span> 定义的 <span class="texhtml" style="white-space: nowrap;">P<span class="t-su t-su-b">m<br />
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l</span></span> 忽略了它。</p>
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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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<li>若 <span class="texhtml" style="white-space: nowrap;">l≥128</span> ，则行为是实现定义的</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>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>球谐函数的一种实现 <a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/1_62_0/libs/math/doc/html/math_toolkit/sf_poly/sph_harm.html">可用于 boost.math</a> ，且它在以设为零的参数 phi 调用时规约到此函数。</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">// 对于 l=3, m=0 的点检查</span>
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<span class="kw4">double</span> x <span class="sy1">=</span> <span class="nu16">1.2345</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">"Y_3^0("</span> <span class="sy1"><<</span> x <span class="sy1"><<</span> <span class="st0">") = "</span> <span class="sy1"><<</span> <a href="cpp-numeric-special_math-sph_legendre.html"><span class="kw1444">std::<span class="me2">sph_legendre</span></span></a><span class="br0">(</span><span class="nu0">3</span>, <span class="nu0">0</span>, x<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">// 准确解</span>
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<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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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"exact solution = "</span>
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<span class="sy1"><<</span> <span class="nu16">0.25</span><span class="sy2">*</span><a href="cpp-numeric-math-sqrt.html"><span class="kw1329">std::<span class="me2">sqrt</span></span></a><span class="br0">(</span><span class="nu0">7</span><span class="sy2">/</span>pi<span class="br0">)</span><span class="sy2">*</span><span class="br0">(</span><span class="nu0">5</span><span class="sy2">*</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-cos.html"><span class="kw1334">std::<span class="me2">cos</span></span></a><span class="br0">(</span>x<span class="br0">)</span>,<span class="nu0">3</span><span class="br0">)</span><span class="sy2">-</span><span class="nu0">3</span><span class="sy2">*</span><a href="cpp-numeric-math-cos.html"><span class="kw1334">std::<span class="me2">cos</span></span></a><span class="br0">(</span>x<span class="br0">)</span><span class="br0">)</span>
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<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">Y_3^0(1.2345) = -0.302387
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exact solution = -0.302387</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/SphericalHarmonic.html">Weisstein, Eric W. “球谐”</a>来自 MathWorld--A Wolfram Web Resource 。</p>
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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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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-special_functions-assoc_legendre.html"><span class="t-lines"><span>assoc_legendre</span><span>assoc_legendref</span><span>assoc_legendrel</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span></span></div>
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</div>
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</td>
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<td>连带勒让德多项式<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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