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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>assoc_legendre, <span style="font-size:0.7em; line-height:130%">std::</span>assoc_legendref, <span style="font-size:0.7em; line-height:130%">std::</span>assoc_legendrel</h1>
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<div id="contentSub"><span class="subpages">&lt; <a href="cpp.html">cpp</a>&lrm; | <a href="cpp-numeric.html">numeric</a>&lrm; | <a href="cpp-numeric-special_functions.html">special functions</a></span></div>
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<div>&#23450;&#20041;&#20110;&#22836;&#25991;&#20214; <code><a href="cpp-header-cmath.html">&lt;cmath&gt;</a></code></div>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> &nbsp; &nbsp; &nbsp;assoc_legendre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br /></span>
<p><span class="kw4">float</span> &nbsp; &nbsp; &nbsp; assoc_legendre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br />
<span class="kw4">long</span> <span class="kw4">double</span> assoc_legendre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br />
<span class="kw4">float</span> &nbsp; &nbsp; &nbsp; assoc_legendref<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">float</span> x <span class="br0">)</span><span class="sy4">;</span><br /></p>
<span class="kw4">long</span> <span class="kw4">double</span> assoc_legendrel<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span></div>
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<td>(1)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> &nbsp; &nbsp; &nbsp;assoc_legendre<span class="br0">(</span> <span class="kw4">unsigned</span> <span class="kw4">int</span> n, <span class="kw4">unsigned</span> <span class="kw4">int</span> m, IntegralType x <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(2)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div class="t-li1"><span class="t-li">1)</span> &#20197;&#21442;&#25968; <code>x</code> &#35745;&#31639; <code>n</code> &#27425; <code>m</code> &#38454;<a href="https://en.wikipedia.org/wiki/Associated_Legendre_polynomials" class="extiw" title="enwiki:Associated Legendre polynomials">&#20851;&#32852;&#21202;&#35753;&#24503;&#22810;&#39033;&#24335;</a></div>
<div class="t-li1"><span class="t-li">2)</span> &#25509;&#21463;&#20219;&#20309;<a href="cpp-types-is_integral.html">&#25972;&#25968;&#31867;&#22411;</a>&#21442;&#25968;&#30340;&#37325;&#36733;&#38598;&#25110;&#20989;&#25968;&#27169;&#26495;&#12290;&#31561;&#20215;&#20110;&#23558;&#21442;&#25968;&#36716;&#22411;&#21040; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> &#21518;&#30340; <span class="t-v">(1)</span> &#12290;</div>
<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">&#21442;&#25968;</span></h3>
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<td>n</td>
<td>-</td>
<td>&#22810;&#39033;&#24335;&#30340;&#27425;&#25968;&#65292;&#26080;&#31526;&#21495;&#25972;&#25968;&#20540;</td>
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<td>m</td>
<td>-</td>
<td>&#22810;&#39033;&#24335;&#30340;&#38454;&#25968;&#65292;&#26080;&#31526;&#21495;&#25972;&#25968;&#20540;</td>
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<td>x</td>
<td>-</td>
<td>&#21442;&#25968;&#65292;&#28014;&#28857;&#25110;&#25972;&#25968;&#31867;&#22411;&#20540;</td>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">&#36820;&#22238;&#20540;</span></h3>
&#33509;&#26080;&#38169;&#35823;&#21457;&#29983;&#65292;&#21017;&#36820;&#22238; <code>x</code> &#30340;&#20851;&#32852;&#21202;&#35753;&#24503;&#22810;&#39033;&#24335; <span class="mjax" style="display:none">\(\mathsf{P}_n^m\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">P<span class="t-su t-su-b">m<br />
n</span></span> &#30340;&#20540;&#65292;&#21363; <span class="mjax" style="display:none">\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">(1-x<span class="t-su">2<br /></span>)<span class="t-su">m/2<br /></span> <span class="t-mfrac"></span></span>
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<td>d<span class="t-su">m<br /></span></td>
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<td>dx<span class="t-su">m<br /></span></td>
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P<span class="t-su t-su-b"><br />
n</span>(x) &#65288;&#20854;&#20013; <span class="mjax" style="display:none">\(\mathsf{P}_n(x)\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">P<span class="t-su t-su-b"><br />
n</span>(x)</span> &#26159;&#38750;&#20851;&#21202;&#35753;&#24503;&#22810;&#39033;&#24335; <span class="t-c"><span class="mw-geshi cpp source-cpp"><a href="cpp-numeric-special_math-legendre.html"><span class="kw1432">std::<span class="me2">legendre</span></span></a><span class="br0">(</span>n, x<span class="br0">)</span></span></span> &#65289;&#12290;
<p>&#27880;&#24847;&#27492;&#23450;&#20041;&#24573;&#30053; <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Condon-ShortleyPhase.html">Condon-Shortley &#30456;&#20301;&#39033;</a> <span class="mjax" style="display:none">\((-1)^m\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">(-1)<span class="t-su">m<br /></span></span> &#12290;</p>
<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">&#38169;&#35823;&#22788;&#29702;</span></h3>
<p>&#21487;&#33021;&#25253;&#21578; <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> &#20013;&#25351;&#23450;&#30340;&#38169;&#35823;&#12290;</p>
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<li>&#33509;&#21442;&#25968;&#26159; NaN &#65292;&#21017;&#36820;&#22238; NaN &#19988;&#19981;&#25253;&#21578;&#23450;&#20041;&#22495;&#38169;&#35823;</li>
<li>&#33509; <span class="texhtml" style="white-space: nowrap;">|x| &gt; 1</span> &#65292;&#21017;&#21487;&#33021;&#20986;&#29616;&#23450;&#20041;&#22495;&#38169;&#35823;</li>
<li>&#33509; <code>n</code> &#22823;&#20110;&#25110;&#31561;&#20110; 128 &#65292;&#21017;&#34892;&#20026;&#26159;&#23454;&#29616;&#23450;&#20041;&#30340;&#12290;</li>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">&#27880;&#35299;</span></h3>
<p>&#19981;&#25903;&#25345; C++17 &#65292;&#20294;&#25903;&#25345; <a href="cpp-experimental-special_functions.html">ISO 29124:2010</a> &#30340;&#23454;&#29616;&#20250;&#25552;&#20379;&#27492;&#20989;&#25968;&#65292;&#33509;&#23454;&#29616;&#23450;&#20041;&#20102; <code>__STDCPP_MATH_SPEC_FUNCS__</code> &#20026;&#33267;&#23569; 201003L &#30340;&#20540;&#65292;&#19988;&#29992;&#25143;&#22312;&#21253;&#21547;&#20219;&#20309;&#26631;&#20934;&#24211;&#22836;&#25991;&#20214;&#21069;&#23450;&#20041;&#20102; <code>__STDCPP_WANT_MATH_SPEC_FUNCS__</code> &#12290;</p>
<p>&#19981;&#25903;&#25345; ISO 29124:2010 &#20294;&#25903;&#25345; TR 19768:2007 (TR1) &#30340;&#23454;&#29616;&#65292;&#22312;&#22836;&#25991;&#20214; <code>tr1/cmath</code> &#21450;&#21629;&#21517;&#31354;&#38388; <code>std::tr1</code> &#20013;&#25552;&#20379;&#27492;&#20989;&#25968;&#12290;</p>
<p>&#27492;&#20989;&#25968;&#30340;&#19968;&#31181;&#23454;&#29616;&#20134;&#20316;&#20026; <code>boost::math::legendre_p</code> <a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/legendre.html">&#21487;&#29992;&#20110; boost.math</a> &#65292;&#38500;&#20102; boost.math &#23450;&#20041;&#21253;&#21547; Condon-Shortley &#30456;&#20301;&#39033;&#12290;</p>
<p>&#21069;&#20960;&#20010;&#20851;&#32852;&#21202;&#35753;&#24503;&#22810;&#39033;&#24335;&#26159;&#65306;</p>
<ul>
<li>assoc_legendre(0, 0, x) = 1</li>
<li>assoc_legendre(1, 0, x) = <span class="texhtml" style="white-space: nowrap;">x</span></li>
<li>assoc_legendre(1, 1, x) = <span class="texhtml" style="white-space: nowrap;">(1-x<span class="t-su">2<br /></span>)<span class="t-su">1/2<br /></span></span></li>
<li>assoc_legendre(2, 0, x) = <span class="texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<td>1</td>
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<td>2</td>
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(3x<span class="t-su">2<br /></span>-1)</li>
<li>assoc_legendre(2, 1, x) = <span class="texhtml" style="white-space: nowrap;">3x(1-x<span class="t-su">2<br /></span>)<span class="t-su">1/2<br /></span></span></li>
<li>assoc_legendre(2, 2, x) = <span class="texhtml" style="white-space: nowrap;">3(1-x<span class="t-su">2<br /></span>)</span></li>
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">&#31034;&#20363;</span></h3>
<div class="t-example">
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<div class="cpp source-cpp">
<pre class="de1"><span class="co2">#include &lt;cmath&gt;</span>
<span class="co2">#include &lt;iostream&gt;</span>
<span class="kw4">double</span> P20<span class="br0">(</span><span class="kw4">double</span> x<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="nu16">0.5</span><span class="sy2">*</span><span class="br0">(</span><span class="nu0">3</span><span class="sy2">*</span>x<span class="sy2">*</span>x<span class="sy2">-</span><span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span> <span class="br0">}</span>
<span class="kw4">double</span> P21<span class="br0">(</span><span class="kw4">double</span> x<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="sy2">-</span><span class="nu16">3.0</span><span class="sy2">*</span>x<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">1</span><span class="sy2">-</span>x<span class="sy2">*</span>x<span class="br0">)</span><span class="sy4">;</span> <span class="br0">}</span>
<span class="kw4">double</span> P22<span class="br0">(</span><span class="kw4">double</span> x<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">return</span> <span class="nu0">3</span><span class="sy2">*</span><span class="br0">(</span><span class="nu0">1</span><span class="sy2">-</span>x<span class="sy2">*</span>x<span class="br0">)</span><span class="sy4">;</span> <span class="br0">}</span>
<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
<span class="br0">{</span>
<span class="co1">// &#28857;&#26816;&#26597;</span>
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1">&lt;&lt;</span> <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><span class="nu0">2</span>, <span class="nu0">0</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;=&#39;</span> <span class="sy1">&lt;&lt;</span> P20<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span>
<span class="sy1">&lt;&lt;</span> <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><span class="nu0">2</span>, <span class="nu0">1</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;=&#39;</span> <span class="sy1">&lt;&lt;</span> P21<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span>
<span class="sy1">&lt;&lt;</span> <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><span class="nu0">2</span>, <span class="nu0">2</span>, <span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;=&#39;</span> <span class="sy1">&lt;&lt;</span> P22<span class="br0">(</span><span class="nu16">0.5</span><span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span><span class="sy4">;</span>
<span class="br0">}</span></pre></div>
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<p>&#36755;&#20986;&#65306;</p>
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<div class="text source-text">
<pre class="de1">-0.125=-0.125
1.29904=1.29904
2.25=2.25</pre></div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">&#21442;&#38405;</span></h3>
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<div><a href="cpp-numeric-special_functions-legendre.html"><span class="t-lines"><span>legendre</span><span>legendref</span><span>legendrel</span></span></a></div>
<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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<td>&#21202;&#35753;&#24503;&#22810;&#39033;&#24335;<br />
<span class="t-mark">(&#20989;&#25968;)</span></td>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">&#22806;&#37096;&#38142;&#25509;</span></h3>
<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html">Weisstein, Eric W. &ldquo;&#20851;&#32852;&#21202;&#35753;&#24503;&#22810;&#39033;&#24335;&rdquo;</a>&#26469;&#33258; MathWorld--A Wolfram Web Resource &#12290;</p>
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