271 lines
16 KiB
HTML
271 lines
16 KiB
HTML
<!DOCTYPE html>
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<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
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<head><meta http-equiv="x-ua-compatible" content="ie=edge">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<title>数学特殊函数</title>
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<link rel="stylesheet" href="ext.css" />
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<link rel="stylesheet" href="site_modules.css" />
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</head>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_special_functions skin-cppreference2 action-view cpp-navbar">
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<div id="cpp-content-base">
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<div id="content"><a id="top"></a>
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<h1 id="firstHeading" class="firstHeading">数学特殊函数</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></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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<p>数学特殊函数库原是库 TR1 ISO/IEC TR 19768:2007 的一部分，后作为独立 ISO 标准， ISO/IEC 29124:2010 出版，最终从 C++17 起并入 ISO C++ 。</p>
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<p>此库的 ISO/IEC 29124:2010 版本可参阅<a href="cpp-experimental-special_functions.html">数学特殊函数</a>。</p>
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<h3><span class="mw-headline" id=".E5.87.BD.E6.95.B0">函数</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc-header">
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<td colspan="2">
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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>
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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-special_functions-assoc_laguerre.html"><span class="t-lines"><span>assoc_laguerre</span><span>assoc_laguerref</span><span>assoc_laguerrel</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>连带拉盖尔多项式<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<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-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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<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-special_functions-beta.html"><span class="t-lines"><span>beta</span><span>betaf</span><span>betal</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>beta 函数<br />
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<span class="t-mark">(函数)</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-special_functions-comp_ellint_1.html"><span class="t-lines"><span>comp_ellint_1</span><span>comp_ellint_1f</span><span>comp_ellint_1l</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>第一类（完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-comp_ellint_2.html"><span class="t-lines"><span>comp_ellint_2</span><span>comp_ellint_2f</span><span>comp_ellint_2l</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>第二类（完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-comp_ellint_3.html"><span class="t-lines"><span>comp_ellint_3</span><span>comp_ellint_3f</span><span>comp_ellint_3l</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>第三类（完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-cyl_bessel_i.html"><span class="t-lines"><span>cyl_bessel_i</span><span>cyl_bessel_if</span><span>cyl_bessel_il</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>规则变形圆柱贝塞尔函数<br />
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<span class="t-mark">(函数)</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-special_functions-cyl_bessel_j.html"><span class="t-lines"><span>cyl_bessel_j</span><span>cyl_bessel_jf</span><span>cyl_bessel_jl</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>（第一类）圆柱贝塞尔函数<br />
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<span class="t-mark">(函数)</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-special_functions-cyl_bessel_k.html"><span class="t-lines"><span>cyl_bessel_k</span><span>cyl_bessel_kf</span><span>cyl_bessel_kl</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>不规则变形圆柱贝塞尔函数<br />
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<span class="t-mark">(函数)</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-special_functions-cyl_neumann.html"><span class="t-lines"><span>cyl_neumann</span><span>cyl_neumannf</span><span>cyl_neumannl</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>圆柱诺依曼函数<br />
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<span class="t-mark">(函数)</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-special_functions-ellint_1.html"><span class="t-lines"><span>ellint_1</span><span>ellint_1f</span><span>ellint_1l</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>第一类（不完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-ellint_2.html"><span class="t-lines"><span>ellint_2</span><span>ellint_2f</span><span>ellint_2l</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>第二类（不完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-ellint_3.html"><span class="t-lines"><span>ellint_3</span><span>ellint_3f</span><span>ellint_3l</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>第三类（不完全）椭圆积分<br />
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<span class="t-mark">(函数)</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-special_functions-expint.html"><span class="t-lines"><span>expint</span><span>expintf</span><span>expintl</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>指数积分<br />
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<span class="t-mark">(函数)</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-special_functions-hermite.html"><span class="t-lines"><span>hermite</span><span>hermitef</span><span>hermitel</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>埃尔米特多项式<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<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-special_functions-legendre.html"><span class="t-lines"><span>legendre</span><span>legendref</span><span>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>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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<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-special_functions-laguerre.html"><span class="t-lines"><span>laguerre</span><span>laguerref</span><span>laguerrel</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>拉盖尔多项式<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<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-special_functions-riemann_zeta.html"><span class="t-lines"><span>riemann_zeta</span><span>riemann_zetaf</span><span>riemann_zetal</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>黎曼 zeta 函数<br />
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<span class="t-mark">(函数)</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-special_functions-sph_bessel.html"><span class="t-lines"><span>sph_bessel</span><span>sph_besself</span><span>sph_bessell</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>（第一类）球面贝塞尔函数<br />
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<span class="t-mark">(函数)</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-special_functions-sph_legendre.html"><span class="t-lines"><span>sph_legendre</span><span>sph_legendref</span><span>sph_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>球面连带勒让德函数<br />
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<span class="t-mark">(函数)</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-special_functions-sph_neumann.html"><span class="t-lines"><span>sph_neumann</span><span>sph_neumannf</span><span>sph_neumannl</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>球面诺依曼函数<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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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">注解</span></h3>
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<p>数学特殊函数的重载在 ISO/IEC 29124:2010 的最终草案（ <a rel="nofollow" class="external text" href="https://wg21.link/N3060">N3060</a> ）中存在，但在 C++17/20 中缺失。有些实现不提供这些重载。我们有意认为这些重载的缺失是 C++17/20 中的缺陷。参阅 <a rel="nofollow" class="external text" href="https://cplusplus.github.io/LWG/issue3234">LWG 问题 3234</a> 。</p>
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<p>这些函数与类类型的<a href="cpp-language-member_functions.html#.E7.89.B9.E6.AE.8A.E6.88.90.E5.91.98.E5.87.BD.E6.95.B0">特殊成员函数</a>无关。</p>
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<h3><span class="mw-headline" id=".E5.BC.95.E7.94.A8">引用</span></h3>
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<div class="t-ref-std-c++20">
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<ul>
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<li>C++20 标准（ISO/IEC 14882:2020）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>26.8.6 Mathematical special functions [sf.cmath]</li>
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</ul>
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</dd>
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</dl>
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</div>
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<div class="t-ref-std-c++17">
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<ul>
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<li>C++17 标准（ISO/IEC 14882:2017）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>29.9.5 Mathematical special functions [sf.cmath]</li>
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</ul>
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</dd>
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</dl>
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</div>
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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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