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Functions</h2></td></tr>
<tr class="memitem:ga377bb7e038c464a27dfe0573fd2d7b33"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga377bb7e038c464a27dfe0573fd2d7b33"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga377bb7e038c464a27dfe0573fd2d7b33">std::assoc_laguerre</a> (unsigned int __n, unsigned int __m, _Tp __x)</td></tr>
<tr class="separator:ga377bb7e038c464a27dfe0573fd2d7b33"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaf83d98f350a1cfcebee6a1f723cf90d2"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaf83d98f350a1cfcebee6a1f723cf90d2">std::assoc_laguerref</a> (unsigned int __n, unsigned int __m, float __x)</td></tr>
<tr class="separator:gaf83d98f350a1cfcebee6a1f723cf90d2"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gac8e245671fb2df5de5fd978d03081f6c"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gac8e245671fb2df5de5fd978d03081f6c">std::assoc_laguerrel</a> (unsigned int __n, unsigned int __m, long double __x)</td></tr>
<tr class="separator:gac8e245671fb2df5de5fd978d03081f6c"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga355349f79119c1fd1e2a9351cec57f0f"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga355349f79119c1fd1e2a9351cec57f0f"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga355349f79119c1fd1e2a9351cec57f0f">std::assoc_legendre</a> (unsigned int __l, unsigned int __m, _Tp __x)</td></tr>
<tr class="separator:ga355349f79119c1fd1e2a9351cec57f0f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga3ced07ddd24bf4af56e2712d148e7f57"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga3ced07ddd24bf4af56e2712d148e7f57">std::assoc_legendref</a> (unsigned int __l, unsigned int __m, float __x)</td></tr>
<tr class="separator:ga3ced07ddd24bf4af56e2712d148e7f57"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga55977b425a539146f060dec1c8003344"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga55977b425a539146f060dec1c8003344">std::assoc_legendrel</a> (unsigned int __l, unsigned int __m, long double __x)</td></tr>
<tr class="separator:ga55977b425a539146f060dec1c8003344"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga6a7220c87c942db48b18b527d92bbd2d"><td class="memTemplParams" colspan="2">template<typename _Tpa , typename _Tpb > </td></tr>
<tr class="memitem:ga6a7220c87c942db48b18b527d92bbd2d"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tpa, _Tpb >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga6a7220c87c942db48b18b527d92bbd2d">std::beta</a> (_Tpa __a, _Tpb __b)</td></tr>
<tr class="separator:ga6a7220c87c942db48b18b527d92bbd2d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga12dc61ee4c09172151cf092ed387e203"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga12dc61ee4c09172151cf092ed387e203">std::betaf</a> (float __a, float __b)</td></tr>
<tr class="separator:ga12dc61ee4c09172151cf092ed387e203"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga8caca1cef099f41a88111209c36ce06c"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga8caca1cef099f41a88111209c36ce06c">std::betal</a> (long double __a, long double __b)</td></tr>
<tr class="separator:ga8caca1cef099f41a88111209c36ce06c"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gac559500c604c43ea943d593c9ad9d289"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:gac559500c604c43ea943d593c9ad9d289"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gac559500c604c43ea943d593c9ad9d289">std::comp_ellint_1</a> (_Tp __k)</td></tr>
<tr class="separator:gac559500c604c43ea943d593c9ad9d289"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga7fb5be999a8125cf7e55e630eb8444a1"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga7fb5be999a8125cf7e55e630eb8444a1">std::comp_ellint_1f</a> (float __k)</td></tr>
<tr class="separator:ga7fb5be999a8125cf7e55e630eb8444a1"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga7247d3dd77c1ff5df3c059fed862dc48"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga7247d3dd77c1ff5df3c059fed862dc48">std::comp_ellint_1l</a> (long double __k)</td></tr>
<tr class="separator:ga7247d3dd77c1ff5df3c059fed862dc48"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga22fcc678829f0daf2de257896378e7e0"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga22fcc678829f0daf2de257896378e7e0"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga22fcc678829f0daf2de257896378e7e0">std::comp_ellint_2</a> (_Tp __k)</td></tr>
<tr class="separator:ga22fcc678829f0daf2de257896378e7e0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga21700f2f125c42b1f1da1f9c7eea1135"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga21700f2f125c42b1f1da1f9c7eea1135">std::comp_ellint_2f</a> (float __k)</td></tr>
<tr class="separator:ga21700f2f125c42b1f1da1f9c7eea1135"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga47b647ec386c8d4b18a030c97842df18"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga47b647ec386c8d4b18a030c97842df18">std::comp_ellint_2l</a> (long double __k)</td></tr>
<tr class="separator:ga47b647ec386c8d4b18a030c97842df18"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gad833404645e24b7f0598a640ff92d623"><td class="memTemplParams" colspan="2">template<typename _Tp , typename _Tpn > </td></tr>
<tr class="memitem:gad833404645e24b7f0598a640ff92d623"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tp, _Tpn >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gad833404645e24b7f0598a640ff92d623">std::comp_ellint_3</a> (_Tp __k, _Tpn __nu)</td></tr>
<tr class="separator:gad833404645e24b7f0598a640ff92d623"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga76834d3112f777703330892303267a39"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga76834d3112f777703330892303267a39">std::comp_ellint_3f</a> (float __k, float __nu)</td></tr>
<tr class="separator:ga76834d3112f777703330892303267a39"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga1ca081fee102cd0d4d6b091285e495e5"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1ca081fee102cd0d4d6b091285e495e5">std::comp_ellint_3l</a> (long double __k, long double __nu)</td></tr>
<tr class="separator:ga1ca081fee102cd0d4d6b091285e495e5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga1c9b5a5c36f000a4f0a55f7fcc486cb0"><td class="memTemplParams" colspan="2">template<typename _Tpnu , typename _Tp > </td></tr>
<tr class="memitem:ga1c9b5a5c36f000a4f0a55f7fcc486cb0"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga1c9b5a5c36f000a4f0a55f7fcc486cb0">std::cyl_bessel_i</a> (_Tpnu __nu, _Tp __x)</td></tr>
<tr class="separator:ga1c9b5a5c36f000a4f0a55f7fcc486cb0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaaf738427d4da0bda66bc2274dfb853a7"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaaf738427d4da0bda66bc2274dfb853a7">std::cyl_bessel_if</a> (float __nu, float __x)</td></tr>
<tr class="separator:gaaf738427d4da0bda66bc2274dfb853a7"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gab7962629216d03efb8ecaa3f70c6878f"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gab7962629216d03efb8ecaa3f70c6878f">std::cyl_bessel_il</a> (long double __nu, long double __x)</td></tr>
<tr class="separator:gab7962629216d03efb8ecaa3f70c6878f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga47e21a13b6d68d0d7f057699bd3b3ce0"><td class="memTemplParams" colspan="2">template<typename _Tpnu , typename _Tp > </td></tr>
<tr class="memitem:ga47e21a13b6d68d0d7f057699bd3b3ce0"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga47e21a13b6d68d0d7f057699bd3b3ce0">std::cyl_bessel_j</a> (_Tpnu __nu, _Tp __x)</td></tr>
<tr class="separator:ga47e21a13b6d68d0d7f057699bd3b3ce0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga15731a7bccd6351d28353e3c4c2a2d23"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga15731a7bccd6351d28353e3c4c2a2d23">std::cyl_bessel_jf</a> (float __nu, float __x)</td></tr>
<tr class="separator:ga15731a7bccd6351d28353e3c4c2a2d23"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gade8e94a80520a8b628b2d658755b25c0"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gade8e94a80520a8b628b2d658755b25c0">std::cyl_bessel_jl</a> (long double __nu, long double __x)</td></tr>
<tr class="separator:gade8e94a80520a8b628b2d658755b25c0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga76dcd3884620955680112aca0d327ada"><td class="memTemplParams" colspan="2">template<typename _Tpnu , typename _Tp > </td></tr>
<tr class="memitem:ga76dcd3884620955680112aca0d327ada"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga76dcd3884620955680112aca0d327ada">std::cyl_bessel_k</a> (_Tpnu __nu, _Tp __x)</td></tr>
<tr class="separator:ga76dcd3884620955680112aca0d327ada"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga1f50047f9aab0ec8b1a1615fe9fbe32f"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1f50047f9aab0ec8b1a1615fe9fbe32f">std::cyl_bessel_kf</a> (float __nu, float __x)</td></tr>
<tr class="separator:ga1f50047f9aab0ec8b1a1615fe9fbe32f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gac35194b926270d7857d651e06198c7d3"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gac35194b926270d7857d651e06198c7d3">std::cyl_bessel_kl</a> (long double __nu, long double __x)</td></tr>
<tr class="separator:gac35194b926270d7857d651e06198c7d3"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga5b7c72ab85e361cbd73f1a3b5f0725a6"><td class="memTemplParams" colspan="2">template<typename _Tpnu , typename _Tp > </td></tr>
<tr class="memitem:ga5b7c72ab85e361cbd73f1a3b5f0725a6"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga5b7c72ab85e361cbd73f1a3b5f0725a6">std::cyl_neumann</a> (_Tpnu __nu, _Tp __x)</td></tr>
<tr class="separator:ga5b7c72ab85e361cbd73f1a3b5f0725a6"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga604c13e8f2bb7cd3c7c91d8b19d6b13a"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga604c13e8f2bb7cd3c7c91d8b19d6b13a">std::cyl_neumannf</a> (float __nu, float __x)</td></tr>
<tr class="separator:ga604c13e8f2bb7cd3c7c91d8b19d6b13a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gaf8986bae9a523c48d861d233835bda8f"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaf8986bae9a523c48d861d233835bda8f">std::cyl_neumannl</a> (long double __nu, long double __x)</td></tr>
<tr class="separator:gaf8986bae9a523c48d861d233835bda8f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gae6b3df5556f38a7d72f9b4457d856f9c"><td class="memTemplParams" colspan="2">template<typename _Tp , typename _Tpp > </td></tr>
<tr class="memitem:gae6b3df5556f38a7d72f9b4457d856f9c"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tp, _Tpp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gae6b3df5556f38a7d72f9b4457d856f9c">std::ellint_1</a> (_Tp __k, _Tpp __phi)</td></tr>
<tr class="separator:gae6b3df5556f38a7d72f9b4457d856f9c"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga308d23d70f4b5e848eb7a4173628ef3b"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga308d23d70f4b5e848eb7a4173628ef3b">std::ellint_1f</a> (float __k, float __phi)</td></tr>
<tr class="separator:ga308d23d70f4b5e848eb7a4173628ef3b"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga795383fa51e02351000b410b478d824f"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga795383fa51e02351000b410b478d824f">std::ellint_1l</a> (long double __k, long double __phi)</td></tr>
<tr class="separator:ga795383fa51e02351000b410b478d824f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:gad6dd71db2b3f90d24ff49bf8cf37bc37"><td class="memTemplParams" colspan="2">template<typename _Tp , typename _Tpp > </td></tr>
<tr class="memitem:gad6dd71db2b3f90d24ff49bf8cf37bc37"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_2< _Tp, _Tpp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gad6dd71db2b3f90d24ff49bf8cf37bc37">std::ellint_2</a> (_Tp __k, _Tpp __phi)</td></tr>
<tr class="separator:gad6dd71db2b3f90d24ff49bf8cf37bc37"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga594a730163c6228c75b152462700062b"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga594a730163c6228c75b152462700062b">std::ellint_2f</a> (float __k, float __phi)</td></tr>
<tr class="separator:ga594a730163c6228c75b152462700062b"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga5c791332d374a809d8ca16c69a1a30f5"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga5c791332d374a809d8ca16c69a1a30f5">std::ellint_2l</a> (long double __k, long double __phi)</td></tr>
<tr class="separator:ga5c791332d374a809d8ca16c69a1a30f5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ga20832e3a67d25cc3d415cafc88019ac3"><td class="memTemplParams" colspan="2">template<typename _Tp , typename _Tpn , typename _Tpp > </td></tr>
<tr class="memitem:ga20832e3a67d25cc3d415cafc88019ac3"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote_3< _Tp, _Tpn, _Tpp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga20832e3a67d25cc3d415cafc88019ac3">std::ellint_3</a> (_Tp __k, _Tpn __nu, _Tpp __phi)</td></tr>
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<tr class="memitem:ga1a80bd2c15bc9fbecda2630a9e9409e7"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1a80bd2c15bc9fbecda2630a9e9409e7">std::ellint_3f</a> (float __k, float __nu, float __phi)</td></tr>
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<tr class="memitem:gaa8c0e5864df8769021a7f3e21a30c5d2"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaa8c0e5864df8769021a7f3e21a30c5d2">std::ellint_3l</a> (long double __k, long double __nu, long double __phi)</td></tr>
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<tr class="memitem:ga88ba17f5d050a6973ca4db1bf6e90590"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga88ba17f5d050a6973ca4db1bf6e90590"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga88ba17f5d050a6973ca4db1bf6e90590">std::expint</a> (_Tp __x)</td></tr>
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<tr class="memitem:ga5842816f6eed2594e0a327df4e4a2a47"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga5842816f6eed2594e0a327df4e4a2a47">std::expintf</a> (float __x)</td></tr>
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<tr class="memitem:ga1329130b32328d0666e290ee5931fa4f"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1329130b32328d0666e290ee5931fa4f">std::expintl</a> (long double __x)</td></tr>
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<tr class="memitem:ga97632b8bf77c323b2369e8d327965bdf"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga97632b8bf77c323b2369e8d327965bdf"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga97632b8bf77c323b2369e8d327965bdf">std::hermite</a> (unsigned int __n, _Tp __x)</td></tr>
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<tr class="memitem:ga94dae7444bb349e33057a92932db8abe"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga94dae7444bb349e33057a92932db8abe">std::hermitef</a> (unsigned int __n, float __x)</td></tr>
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<tr class="memitem:ga21f8e312ee3e65286f86bf141b0f32e0"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga21f8e312ee3e65286f86bf141b0f32e0">std::hermitel</a> (unsigned int __n, long double __x)</td></tr>
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<tr class="memitem:gacae65579b397fddcd27954380d364a58"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:gacae65579b397fddcd27954380d364a58"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gacae65579b397fddcd27954380d364a58">std::laguerre</a> (unsigned int __n, _Tp __x)</td></tr>
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<tr class="memitem:gada763419b0e21b38e38daa8b6eb56a8c"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gada763419b0e21b38e38daa8b6eb56a8c">std::laguerref</a> (unsigned int __n, float __x)</td></tr>
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<tr class="memitem:gaaf8b141edf9163b37ea4f2ed3e0191fc"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaaf8b141edf9163b37ea4f2ed3e0191fc">std::laguerrel</a> (unsigned int __n, long double __x)</td></tr>
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<tr class="memitem:gaf6eac7fcb98e25b8f3f7d1b95fa7add8"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:gaf6eac7fcb98e25b8f3f7d1b95fa7add8"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#gaf6eac7fcb98e25b8f3f7d1b95fa7add8">std::legendre</a> (unsigned int __l, _Tp __x)</td></tr>
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<tr class="memitem:gaed94e3c664c99f5204da75be75aeac21"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaed94e3c664c99f5204da75be75aeac21">std::legendref</a> (unsigned int __l, float __x)</td></tr>
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<tr class="memitem:ga1b39bc22e3cc4860d08eb54099460391"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1b39bc22e3cc4860d08eb54099460391">std::legendrel</a> (unsigned int __l, long double __x)</td></tr>
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<tr class="memitem:ga67a6bfed9b6ab692e8c798b674431424"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga67a6bfed9b6ab692e8c798b674431424">std::riemann_zeta</a> (_Tp __s)</td></tr>
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<tr class="memitem:gaf92063315061a56d3e2c4053156d968e"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaf92063315061a56d3e2c4053156d968e">std::riemann_zetaf</a> (float __s)</td></tr>
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<tr class="memitem:ga1e92da3b878d75270f38d3ec9b513086"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga1e92da3b878d75270f38d3ec9b513086">std::riemann_zetal</a> (long double __s)</td></tr>
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<tr class="memitem:ga478e517ed975bcb256de230e64f0fda5"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga478e517ed975bcb256de230e64f0fda5"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga478e517ed975bcb256de230e64f0fda5">std::sph_bessel</a> (unsigned int __n, _Tp __x)</td></tr>
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<tr class="memitem:ga534e36e1dcefad8daec98920db16eec4"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga534e36e1dcefad8daec98920db16eec4">std::sph_besself</a> (unsigned int __n, float __x)</td></tr>
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<tr class="memitem:ga11d72b1af81ce9da3c878a25087ee927"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga11d72b1af81ce9da3c878a25087ee927">std::sph_bessell</a> (unsigned int __n, long double __x)</td></tr>
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<tr class="memitem:ga573842c12247b87746b548f1945755a8"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga573842c12247b87746b548f1945755a8"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga573842c12247b87746b548f1945755a8">std::sph_legendre</a> (unsigned int __l, unsigned int __m, _Tp __theta)</td></tr>
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<tr class="memitem:gaae635d28c06a3be2679901b382090852"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#gaae635d28c06a3be2679901b382090852">std::sph_legendref</a> (unsigned int __l, unsigned int __m, float __theta)</td></tr>
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<tr class="memitem:ga2f6618dea1847f09fd67f3c974c1910d"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga2f6618dea1847f09fd67f3c974c1910d">std::sph_legendrel</a> (unsigned int __l, unsigned int __m, long double __theta)</td></tr>
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<tr class="memitem:ga1cf4362a67ab5bae9e6b69c867e85371"><td class="memTemplParams" colspan="2">template<typename _Tp > </td></tr>
<tr class="memitem:ga1cf4362a67ab5bae9e6b69c867e85371"><td class="memTemplItemLeft" align="right" valign="top">__gnu_cxx::__promote< _Tp >::__type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="a01497.html#ga1cf4362a67ab5bae9e6b69c867e85371">std::sph_neumann</a> (unsigned int __n, _Tp __x)</td></tr>
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<tr class="memitem:ga789143122fa99536329bc2d1b1aac2f0"><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga789143122fa99536329bc2d1b1aac2f0">std::sph_neumannf</a> (unsigned int __n, float __x)</td></tr>
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<tr class="memitem:ga3cededa9b6e4601f190c3811e6aabfd6"><td class="memItemLeft" align="right" valign="top">long double </td><td class="memItemRight" valign="bottom"><a class="el" href="a01497.html#ga3cededa9b6e4601f190c3811e6aabfd6">std::sph_neumannl</a> (unsigned int __n, long double __x)</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<p>A collection of advanced mathematical special functions, defined by ISO/IEC IS 29124. </p>
<h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#ga377bb7e038c464a27dfe0573fd2d7b33">◆ </a></span>assoc_laguerre()</h2>
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<p>Return the associated Laguerre polynomial of nonnegative order <code>n</code>, nonnegative degree <code>m</code> and real argument <code>x:</code> <img class="formulaInl" alt="$ L_n^m(x) $" src="form_41.png"/>.</p>
<p>The associated Laguerre function of real degree <img class="formulaInl" alt="$ \alpha $" src="form_42.png"/>, <img class="formulaInl" alt="$ L_n^\alpha(x) $" src="form_43.png"/>, is defined by </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} {}_1F_1(-n; \alpha + 1; x) \]" src="form_44.png"/>
</p>
<p> where <img class="formulaInl" alt="$ (\alpha)_n $" src="form_45.png"/> is the Pochhammer symbol and <img class="formulaInl" alt="$ {}_1F_1(a; c; x) $" src="form_46.png"/> is the confluent hypergeometric function.</p>
<p>The associated Laguerre polynomial is defined for integral degree <img class="formulaInl" alt="$ \alpha = m $" src="form_47.png"/> by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) \]" src="form_48.png"/>
</p>
<p> where the Laguerre polynomial is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]" src="form_49.png"/>
</p>
<p> and <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>. </p><dl class="section see"><dt>See also</dt><dd>laguerre for details of the Laguerre function of degree <code>n</code> </dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
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</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__n</td><td>The order of the Laguerre function, <code>__n >= 0</code>. </td></tr>
<tr><td class="paramname">__m</td><td>The degree of the Laguerre function, <code>__m >= 0</code>. </td></tr>
<tr><td class="paramname">__x</td><td>The argument of the Laguerre function, <code>__x >= 0</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code>__x < 0</code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00252">252</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaf83d98f350a1cfcebee6a1f723cf90d2">◆ </a></span>assoc_laguerref()</h2>
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<p>Return the associated Laguerre polynomial of order <code>n</code>, degree <code>m:</code> <img class="formulaInl" alt="$ L_n^m(x) $" src="form_41.png"/> for <code>float</code> argument.</p>
<dl class="section see"><dt>See also</dt><dd>assoc_laguerre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00206">206</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gac8e245671fb2df5de5fd978d03081f6c">◆ </a></span>assoc_laguerrel()</h2>
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<td class="paramname"><em>__m</em>, </td>
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<td class="paramname"><em>__x</em> </td>
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<p>Return the associated Laguerre polynomial of order <code>n</code>, degree <code>m:</code> <img class="formulaInl" alt="$ L_n^m(x) $" src="form_41.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>assoc_laguerre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00216">216</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<td>(</td>
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<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__m</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Return the associated Legendre function of degree <code>l</code> and order <code>m</code>.</p>
<p>The associated Legendre function is derived from the Legendre function <img class="formulaInl" alt="$ P_l(x) $" src="form_51.png"/> by the Rodrigues formula: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) \]" src="form_52.png"/>
</p>
<dl class="section see"><dt>See also</dt><dd>legendre for details of the Legendre function of degree <code>l</code> </dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__l</td><td>The degree <code>__l >= 0</code>. </td></tr>
<tr><td class="paramname">__m</td><td>The order <code>__m <= l</code>. </td></tr>
<tr><td class="paramname">__x</td><td>The argument, <code>abs(__x) <= 1</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code>abs(__x) > 1</code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00298">298</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga3ced07ddd24bf4af56e2712d148e7f57">◆ </a></span>assoc_legendref()</h2>
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<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
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<td class="paramname"><em>__m</em>, </td>
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<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
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<p>Return the associated Legendre function of degree <code>l</code> and order <code>m</code> for <code>float</code> argument.</p>
<dl class="section see"><dt>See also</dt><dd>assoc_legendre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00267">267</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga55977b425a539146f060dec1c8003344">◆ </a></span>assoc_legendrel()</h2>
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<td class="memname">long double std::assoc_legendrel </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__m</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
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<td>)</td>
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<p>Return the associated Legendre function of degree <code>l</code> and order <code>m</code>.</p>
<dl class="section see"><dt>See also</dt><dd>assoc_legendre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00276">276</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga6a7220c87c942db48b18b527d92bbd2d">◆ </a></span>beta()</h2>
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template<typename _Tpa , typename _Tpb > </div>
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<td>(</td>
<td class="paramtype">_Tpa </td>
<td class="paramname"><em>__a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tpb </td>
<td class="paramname"><em>__b</em> </td>
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<p>Return the beta function, <img class="formulaInl" alt="$B(a,b)$" src="form_54.png"/>, for real parameters <code>a</code>, <code>b</code>.</p>
<p>The beta function is defined by </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \]" src="form_55.png"/>
</p>
<p> where <img class="formulaInl" alt="$ a > 0 $" src="form_56.png"/> and <img class="formulaInl" alt="$ b > 0 $" src="form_57.png"/></p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tpa</td><td>The floating-point type of the parameter <code>__a</code>. </td></tr>
<tr><td class="paramname">_Tpb</td><td>The floating-point type of the parameter <code>__b</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__a</td><td>The first argument of the beta function, <code> __a > 0 </code>. </td></tr>
<tr><td class="paramname">__b</td><td>The second argument of the beta function, <code> __b > 0 </code>. </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __a < 0 </code> or <code> __b < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00343">343</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga12dc61ee4c09172151cf092ed387e203">◆ </a></span>betaf()</h2>
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<td class="memname">float std::betaf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__b</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<p>Return the beta function, <img class="formulaInl" alt="$ B(a,b) $" src="form_53.png"/>, for <code>float</code> parameters <code>a</code>, <code>b</code>.</p>
<dl class="section see"><dt>See also</dt><dd>beta for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00312">312</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga8caca1cef099f41a88111209c36ce06c">◆ </a></span>betal()</h2>
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<td class="memname">long double std::betal </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__b</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Return the beta function, <img class="formulaInl" alt="$B(a,b)$" src="form_54.png"/>, for long double parameters <code>a</code>, <code>b</code>.</p>
<dl class="section see"><dt>See also</dt><dd>beta for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00322">322</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gac559500c604c43ea943d593c9ad9d289">◆ </a></span>comp_ellint_1()</h2>
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template<typename _Tp > </div>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::comp_ellint_1 </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__k</em></td><td>)</td>
<td></td>
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<p>Return the complete elliptic integral of the first kind <img class="formulaInl" alt="$ K(k) $" src="form_59.png"/> for real modulus <code>k</code>.</p>
<p>The complete elliptic integral of the first kind is defined as </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \]" src="form_60.png"/>
</p>
<p> where <img class="formulaInl" alt="$ F(k,\phi) $" src="form_61.png"/> is the incomplete elliptic integral of the first kind and the modulus <img class="formulaInl" alt="$ |k| <= 1 $" src="form_62.png"/>. </p><dl class="section see"><dt>See also</dt><dd>ellint_1 for details of the incomplete elliptic function of the first kind.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code> abs(__k) <= 1 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> abs(__k) > 1 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00391">391</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga7fb5be999a8125cf7e55e630eb8444a1">◆ </a></span>comp_ellint_1f()</h2>
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<td class="memname">float std::comp_ellint_1f </td>
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<td class="paramtype">float </td>
<td class="paramname"><em>__k</em></td><td>)</td>
<td></td>
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<p>Return the complete elliptic integral of the first kind <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/> for <code>float</code> modulus <code>k</code>.</p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_1 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00358">358</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga7247d3dd77c1ff5df3c059fed862dc48">◆ </a></span>comp_ellint_1l()</h2>
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<td class="memname">long double std::comp_ellint_1l </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em></td><td>)</td>
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<p>Return the complete elliptic integral of the first kind <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/> for long double modulus <code>k</code>.</p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_1 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00368">368</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga22fcc678829f0daf2de257896378e7e0">◆ </a></span>comp_ellint_2()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::comp_ellint_2 </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__k</em></td><td>)</td>
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<p>Return the complete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/> for real modulus <code>k</code>.</p>
<p>The complete elliptic integral of the second kind is defined as </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} \]" src="form_63.png"/>
</p>
<p> where <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/> is the incomplete elliptic integral of the second kind and the modulus <img class="formulaInl" alt="$ |k| <= 1 $" src="form_62.png"/>. </p><dl class="section see"><dt>See also</dt><dd>ellint_2 for details of the incomplete elliptic function of the second kind.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code>abs(__k)</code> <= 1 </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code>abs(__k)</code> > 1. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00438">438</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga21700f2f125c42b1f1da1f9c7eea1135">◆ </a></span>comp_ellint_2f()</h2>
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<td class="paramtype">float </td>
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<p>Return the complete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/> for <code>float</code> modulus <code>k</code>.</p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_2 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00406">406</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga47b647ec386c8d4b18a030c97842df18">◆ </a></span>comp_ellint_2l()</h2>
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<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em></td><td>)</td>
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<p>Return the complete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/> for long double modulus <code>k</code>.</p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_2 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00416">416</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gad833404645e24b7f0598a640ff92d623">◆ </a></span>comp_ellint_3()</h2>
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<td>(</td>
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<td class="paramname"><em>__k</em>, </td>
</tr>
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<td class="paramkey"></td>
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<td class="paramname"><em>__nu</em> </td>
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<p>Return the complete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) $" src="form_66.png"/> for real modulus <code>k</code>.</p>
<p>The complete elliptic integral of the third kind is defined as </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} \frac{d\theta} {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} \]" src="form_67.png"/>
</p>
<p> where <img class="formulaInl" alt="$ \Pi(k,\nu,\phi) $" src="form_68.png"/> is the incomplete elliptic integral of the second kind and the modulus <img class="formulaInl" alt="$ |k| <= 1 $" src="form_62.png"/>. </p><dl class="section see"><dt>See also</dt><dd>ellint_3 for details of the incomplete elliptic function of the third kind.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
<tr><td class="paramname">_Tpn</td><td>The floating-point type of the argument <code>__nu</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code>abs(__k)</code> <= 1 </td></tr>
<tr><td class="paramname">__nu</td><td>The argument </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code>abs(__k)</code> > 1. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00489">489</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga76834d3112f777703330892303267a39">◆ </a></span>comp_ellint_3f()</h2>
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<td class="memname">float std::comp_ellint_3f </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>Return the complete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu) $" src="form_65.png"/> for <code>float</code> modulus <code>k</code>. </p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_3 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00453">453</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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</div>
<a id="ga1ca081fee102cd0d4d6b091285e495e5"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga1ca081fee102cd0d4d6b091285e495e5">◆ </a></span>comp_ellint_3l()</h2>
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<td class="memname">long double std::comp_ellint_3l </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
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</div><div class="memdoc">
<p>Return the complete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu) $" src="form_65.png"/> for <code>long double</code> modulus <code>k</code>. </p>
<dl class="section see"><dt>See also</dt><dd>comp_ellint_3 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00463">463</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<a id="ga1c9b5a5c36f000a4f0a55f7fcc486cb0"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga1c9b5a5c36f000a4f0a55f7fcc486cb0">◆ </a></span>cyl_bessel_i()</h2>
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<div class="memproto">
<div class="memtemplate">
template<typename _Tpnu , typename _Tp > </div>
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<tr>
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<table class="memname">
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<td class="memname">__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_i </td>
<td>(</td>
<td class="paramtype">_Tpnu </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the regular modified Bessel function <img class="formulaInl" alt="$ I_{\nu}(x) $" src="form_69.png"/> for real order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The regular modified cylindrical Bessel function is: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]" src="form_71.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tpnu</td><td>The floating-point type of the order <code>__nu</code>. </td></tr>
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__nu</td><td>The order </td></tr>
<tr><td class="paramname">__x</td><td>The argument, <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00535">535</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaaf738427d4da0bda66bc2274dfb853a7">◆ </a></span>cyl_bessel_if()</h2>
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<td class="memname">float std::cyl_bessel_if </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>Return the regular modified Bessel function <img class="formulaInl" alt="$ I_{\nu}(x) $" src="form_69.png"/> for <code>float</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_i for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00504">504</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<a id="gab7962629216d03efb8ecaa3f70c6878f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gab7962629216d03efb8ecaa3f70c6878f">◆ </a></span>cyl_bessel_il()</h2>
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<td class="memname">long double std::cyl_bessel_il </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
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</div><div class="memdoc">
<p>Return the regular modified Bessel function <img class="formulaInl" alt="$ I_{\nu}(x) $" src="form_69.png"/> for <code>long double</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_i for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00514">514</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga47e21a13b6d68d0d7f057699bd3b3ce0">◆ </a></span>cyl_bessel_j()</h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template<typename _Tpnu , typename _Tp > </div>
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<tr>
<td class="mlabels-left">
<table class="memname">
<tr>
<td class="memname">__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_j </td>
<td>(</td>
<td class="paramtype">_Tpnu </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the Bessel function <img class="formulaInl" alt="$ J_{\nu}(x) $" src="form_72.png"/> of real order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The cylindrical Bessel function is: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} \]" src="form_73.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tpnu</td><td>The floating-point type of the order <code>__nu</code>. </td></tr>
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__nu</td><td>The order </td></tr>
<tr><td class="paramname">__x</td><td>The argument, <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00581">581</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga15731a7bccd6351d28353e3c4c2a2d23">◆ </a></span>cyl_bessel_jf()</h2>
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<td class="memname">float std::cyl_bessel_jf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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</div><div class="memdoc">
<p>Return the Bessel function of the first kind <img class="formulaInl" alt="$ J_{\nu}(x) $" src="form_72.png"/> for <code>float</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_j for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00550">550</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
</div>
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<h2 class="memtitle"><span class="permalink"><a href="#gade8e94a80520a8b628b2d658755b25c0">◆ </a></span>cyl_bessel_jl()</h2>
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<td class="memname">long double std::cyl_bessel_jl </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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</div><div class="memdoc">
<p>Return the Bessel function of the first kind <img class="formulaInl" alt="$ J_{\nu}(x) $" src="form_72.png"/> for <code>long double</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_j for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00560">560</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga76dcd3884620955680112aca0d327ada">◆ </a></span>cyl_bessel_k()</h2>
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<div class="memproto">
<div class="memtemplate">
template<typename _Tpnu , typename _Tp > </div>
<table class="mlabels">
<tr>
<td class="mlabels-left">
<table class="memname">
<tr>
<td class="memname">__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_bessel_k </td>
<td>(</td>
<td class="paramtype">_Tpnu </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the irregular modified Bessel function <img class="formulaInl" alt="$ K_{\nu}(x) $" src="form_74.png"/> of real order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x $" src="form_75.png"/>.</p>
<p>The irregular modified Bessel function is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} \]" src="form_76.png"/>
</p>
<p> where for integral <img class="formulaInl" alt="$ \nu = n $" src="form_77.png"/> a limit is taken: <img class="formulaInl" alt="$ lim_{\nu \to n} $" src="form_78.png"/>. For negative argument we have simply: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ K_{-\nu}(x) = K_{\nu}(x) \]" src="form_79.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tpnu</td><td>The floating-point type of the order <code>__nu</code>. </td></tr>
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__nu</td><td>The order </td></tr>
<tr><td class="paramname">__x</td><td>The argument, <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00633">633</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1f50047f9aab0ec8b1a1615fe9fbe32f">◆ </a></span>cyl_bessel_kf()</h2>
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<td class="memname">float std::cyl_bessel_kf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
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</div><div class="memdoc">
<p>Return the irregular modified Bessel function <img class="formulaInl" alt="$ K_{\nu}(x) $" src="form_74.png"/> for <code>float</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_k for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00596">596</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
</div>
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<h2 class="memtitle"><span class="permalink"><a href="#gac35194b926270d7857d651e06198c7d3">◆ </a></span>cyl_bessel_kl()</h2>
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<td class="memname">long double std::cyl_bessel_kl </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span></span> </td>
</tr>
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<p>Return the irregular modified Bessel function <img class="formulaInl" alt="$ K_{\nu}(x) $" src="form_74.png"/> for <code>long double</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_bessel_k for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00606">606</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga5b7c72ab85e361cbd73f1a3b5f0725a6">◆ </a></span>cyl_neumann()</h2>
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template<typename _Tpnu , typename _Tp > </div>
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<td class="memname">__gnu_cxx::__promote_2<_Tpnu, _Tp>::__type std::cyl_neumann </td>
<td>(</td>
<td class="paramtype">_Tpnu </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Return the Neumann function <img class="formulaInl" alt="$ N_{\nu}(x) $" src="form_80.png"/> of real order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The Neumann function is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} {\sin \nu\pi} \]" src="form_81.png"/>
</p>
<p> where <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/> and for integral order <img class="formulaInl" alt="$ \nu = n $" src="form_77.png"/> a limit is taken: <img class="formulaInl" alt="$ lim_{\nu \to n} $" src="form_78.png"/>.</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tpnu</td><td>The floating-point type of the order <code>__nu</code>. </td></tr>
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__nu</td><td>The order </td></tr>
<tr><td class="paramname">__x</td><td>The argument, <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00681">681</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga604c13e8f2bb7cd3c7c91d8b19d6b13a">◆ </a></span>cyl_neumannf()</h2>
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<td class="memname">float std::cyl_neumannf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the Neumann function <img class="formulaInl" alt="$ N_{\nu}(x) $" src="form_80.png"/> of <code>float</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x $" src="form_75.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_neumann for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00648">648</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaf8986bae9a523c48d861d233835bda8f">◆ </a></span>cyl_neumannl()</h2>
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<td class="memname">long double std::cyl_neumannl </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the Neumann function <img class="formulaInl" alt="$ N_{\nu}(x) $" src="form_80.png"/> of <code>long double</code> order <img class="formulaInl" alt="$ \nu $" src="form_70.png"/> and argument <img class="formulaInl" alt="$ x $" src="form_75.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>cyl_neumann for setails. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00658">658</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gae6b3df5556f38a7d72f9b4457d856f9c">◆ </a></span>ellint_1()</h2>
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<td class="memname">__gnu_cxx::__promote_2<_Tp, _Tpp>::__type std::ellint_1 </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tpp </td>
<td class="paramname"><em>__phi</em> </td>
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<p>Return the incomplete elliptic integral of the first kind <img class="formulaInl" alt="$ F(k,\phi) $" src="form_61.png"/> for <code>real</code> modulus <img class="formulaInl" alt="$ k $" src="form_82.png"/> and angle <img class="formulaInl" alt="$ \phi $" src="form_83.png"/>.</p>
<p>The incomplete elliptic integral of the first kind is defined as </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ F(k,\phi) = \int_0^{\phi}\frac{d\theta} {\sqrt{1 - k^2 sin^2\theta}} \]" src="form_84.png"/>
</p>
<p> For <img class="formulaInl" alt="$ \phi= \pi/2 $" src="form_85.png"/> this becomes the complete elliptic integral of the first kind, <img class="formulaInl" alt="$ K(k) $" src="form_59.png"/>. </p><dl class="section see"><dt>See also</dt><dd>comp_ellint_1.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
<tr><td class="paramname">_Tpp</td><td>The floating-point type of the angle <code>__phi</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code> abs(__k) <= 1 </code> </td></tr>
<tr><td class="paramname">__phi</td><td>The integral limit argument in radians </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> abs(__k) > 1 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00729">729</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga308d23d70f4b5e848eb7a4173628ef3b">◆ </a></span>ellint_1f()</h2>
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<td class="memname">float std::ellint_1f </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the incomplete elliptic integral of the first kind <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/> for <code>float</code> modulus <img class="formulaInl" alt="$ k $" src="form_82.png"/> and angle <img class="formulaInl" alt="$ \phi $" src="form_83.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>ellint_1 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00696">696</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga795383fa51e02351000b410b478d824f">◆ </a></span>ellint_1l()</h2>
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<td class="memname">long double std::ellint_1l </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<span class="mlabels"><span class="mlabel">inline</span></span> </td>
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<p>Return the incomplete elliptic integral of the first kind <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/> for <code>long double</code> modulus <img class="formulaInl" alt="$ k $" src="form_82.png"/> and angle <img class="formulaInl" alt="$ \phi $" src="form_83.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>ellint_1 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00706">706</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gad6dd71db2b3f90d24ff49bf8cf37bc37">◆ </a></span>ellint_2()</h2>
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<td class="memname">__gnu_cxx::__promote_2<_Tp, _Tpp>::__type std::ellint_2 </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tpp </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
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<p>Return the incomplete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/>.</p>
<p>The incomplete elliptic integral of the second kind is defined as </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} \]" src="form_86.png"/>
</p>
<p> For <img class="formulaInl" alt="$ \phi= \pi/2 $" src="form_85.png"/> this becomes the complete elliptic integral of the second kind, <img class="formulaInl" alt="$ E(k) $" src="form_58.png"/>. </p><dl class="section see"><dt>See also</dt><dd>comp_ellint_2.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
<tr><td class="paramname">_Tpp</td><td>The floating-point type of the angle <code>__phi</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code> abs(__k) <= 1 </code> </td></tr>
<tr><td class="paramname">__phi</td><td>The integral limit argument in radians </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The elliptic function of the second kind. </dd></dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> abs(__k) > 1 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00777">777</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga594a730163c6228c75b152462700062b">◆ </a></span>ellint_2f()</h2>
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<td class="memname">float std::ellint_2f </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Return the incomplete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/> for <code>float</code> argument. </p>
<dl class="section see"><dt>See also</dt><dd>ellint_2 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00744">744</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga5c791332d374a809d8ca16c69a1a30f5">◆ </a></span>ellint_2l()</h2>
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<td class="memname">long double std::ellint_2l </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Return the incomplete elliptic integral of the second kind <img class="formulaInl" alt="$ E(k,\phi) $" src="form_64.png"/>. </p>
<dl class="section see"><dt>See also</dt><dd>ellint_2 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00754">754</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga20832e3a67d25cc3d415cafc88019ac3">◆ </a></span>ellint_3()</h2>
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<td class="memname">__gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type std::ellint_3 </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tpn </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tpp </td>
<td class="paramname"><em>__phi</em> </td>
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<p>Return the incomplete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu,\phi) $" src="form_68.png"/>. </p>
<p>The incomplete elliptic integral of the third kind is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \Pi(k,\nu,\phi) = \int_0^{\phi} \frac{d\theta} {(1 - \nu \sin^2\theta) \sqrt{1 - k^2 \sin^2\theta}} \]" src="form_87.png"/>
</p>
<p> For <img class="formulaInl" alt="$ \phi= \pi/2 $" src="form_85.png"/> this becomes the complete elliptic integral of the third kind, <img class="formulaInl" alt="$ \Pi(k,\nu) $" src="form_65.png"/>. </p><dl class="section see"><dt>See also</dt><dd>comp_ellint_3.</dd></dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the modulus <code>__k</code>. </td></tr>
<tr><td class="paramname">_Tpn</td><td>The floating-point type of the argument <code>__nu</code>. </td></tr>
<tr><td class="paramname">_Tpp</td><td>The floating-point type of the angle <code>__phi</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__k</td><td>The modulus, <code> abs(__k) <= 1 </code> </td></tr>
<tr><td class="paramname">__nu</td><td>The second argument </td></tr>
<tr><td class="paramname">__phi</td><td>The integral limit argument in radians </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The elliptic function of the third kind. </dd></dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> abs(__k) > 1 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00830">830</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1a80bd2c15bc9fbecda2630a9e9409e7">◆ </a></span>ellint_3f()</h2>
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<td class="memname">float std::ellint_3f </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__k</em>, </td>
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<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__phi</em> </td>
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<p>Return the incomplete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu,\phi) $" src="form_68.png"/> for <code>float</code> argument. </p>
<dl class="section see"><dt>See also</dt><dd>ellint_3 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00792">792</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaa8c0e5864df8769021a7f3e21a30c5d2">◆ </a></span>ellint_3l()</h2>
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<td class="memname">long double std::ellint_3l </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__k</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__nu</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__phi</em> </td>
</tr>
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<td></td>
<td>)</td>
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<p>Return the incomplete elliptic integral of the third kind <img class="formulaInl" alt="$ \Pi(k,\nu,\phi) $" src="form_68.png"/>. </p>
<dl class="section see"><dt>See also</dt><dd>ellint_3 for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00802">802</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga88ba17f5d050a6973ca4db1bf6e90590">◆ </a></span>expint()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::expint </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em></td><td>)</td>
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<p>Return the exponential integral <img class="formulaInl" alt="$ Ei(x) $" src="form_88.png"/> for <code>real</code> argument <code>x</code>.</p>
<p>The exponential integral is given by </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt \]" src="form_89.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__x</td><td>The argument of the exponential integral function. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00870">870</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga5842816f6eed2594e0a327df4e4a2a47">◆ </a></span>expintf()</h2>
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<td class="memname">float std::expintf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em></td><td>)</td>
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<p>Return the exponential integral <img class="formulaInl" alt="$ Ei(x) $" src="form_88.png"/> for <code>float</code> argument <code>x</code>.</p>
<dl class="section see"><dt>See also</dt><dd>expint for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00844">844</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1329130b32328d0666e290ee5931fa4f">◆ </a></span>expintl()</h2>
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<td class="memname">long double std::expintl </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em></td><td>)</td>
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<p>Return the exponential integral <img class="formulaInl" alt="$ Ei(x) $" src="form_88.png"/> for <code>long double</code> argument <code>x</code>.</p>
<dl class="section see"><dt>See also</dt><dd>expint for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00854">854</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga97632b8bf77c323b2369e8d327965bdf">◆ </a></span>hermite()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::hermite </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
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<td></td>
<td>)</td>
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<p>Return the Hermite polynomial <img class="formulaInl" alt="$ H_n(x) $" src="form_90.png"/> of order n and <code>real</code> argument <code>x</code>.</p>
<p>The Hermite polynomial is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \]" src="form_91.png"/>
</p>
<p>The Hermite polynomial obeys a reflection formula: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ H_n(-x) = (-1)^n H_n(x) \]" src="form_92.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__n</td><td>The order </td></tr>
<tr><td class="paramname">__x</td><td>The argument </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00918">918</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga94dae7444bb349e33057a92932db8abe">◆ </a></span>hermitef()</h2>
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<td class="memname">float std::hermitef </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<p>Return the Hermite polynomial <img class="formulaInl" alt="$ H_n(x) $" src="form_90.png"/> of nonnegative order n and float argument <code>x</code>.</p>
<dl class="section see"><dt>See also</dt><dd>hermite for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00885">885</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga21f8e312ee3e65286f86bf141b0f32e0">◆ </a></span>hermitel()</h2>
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<td class="memname">long double std::hermitel </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
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<p>Return the Hermite polynomial <img class="formulaInl" alt="$ H_n(x) $" src="form_90.png"/> of nonnegative order n and <code>long double</code> argument <code>x</code>.</p>
<dl class="section see"><dt>See also</dt><dd>hermite for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00895">895</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gacae65579b397fddcd27954380d364a58">◆ </a></span>laguerre()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::laguerre </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
</tr>
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<td>)</td>
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<p>Returns the Laguerre polynomial <img class="formulaInl" alt="$ L_n(x) $" src="form_93.png"/> of nonnegative degree <code>n</code> and real argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The Laguerre polynomial is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) \]" src="form_49.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__n</td><td>The nonnegative order </td></tr>
<tr><td class="paramname">__x</td><td>The argument <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00962">962</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gada763419b0e21b38e38daa8b6eb56a8c">◆ </a></span>laguerref()</h2>
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<td class="memname">float std::laguerref </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
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<p>Returns the Laguerre polynomial <img class="formulaInl" alt="$ L_n(x) $" src="form_93.png"/> of nonnegative degree <code>n</code> and <code>float</code> argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>laguerre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00933">933</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaaf8b141edf9163b37ea4f2ed3e0191fc">◆ </a></span>laguerrel()</h2>
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<td class="memname">long double std::laguerrel </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
</tr>
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<td></td>
<td>)</td>
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<p>Returns the Laguerre polynomial <img class="formulaInl" alt="$ L_n(x) $" src="form_93.png"/> of nonnegative degree <code>n</code> and <code>long double</code> argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>laguerre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00943">943</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaf6eac7fcb98e25b8f3f7d1b95fa7add8">◆ </a></span>legendre()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::legendre </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__x</em> </td>
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<p>Return the Legendre polynomial <img class="formulaInl" alt="$ P_l(x) $" src="form_51.png"/> of nonnegative degree <img class="formulaInl" alt="$ l $" src="form_94.png"/> and real argument <img class="formulaInl" alt="$ |x| <= 0 $" src="form_95.png"/>.</p>
<p>The Legendre function of order <img class="formulaInl" alt="$ l $" src="form_94.png"/> and argument <img class="formulaInl" alt="$ x $" src="form_75.png"/>, <img class="formulaInl" alt="$ P_l(x) $" src="form_51.png"/>, is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} \]" src="form_96.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__l</td><td>The degree <img class="formulaInl" alt="$ l >= 0 $" src="form_97.png"/> </td></tr>
<tr><td class="paramname">__x</td><td>The argument <code>abs(__x)</code> <= 1 </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code>abs(__x)</code> > 1 </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01007">1007</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaed94e3c664c99f5204da75be75aeac21">◆ </a></span>legendref()</h2>
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<td class="memname">float std::legendref </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
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<td class="paramname"><em>__x</em> </td>
</tr>
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<td></td>
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<p>Return the Legendre polynomial <img class="formulaInl" alt="$ P_l(x) $" src="form_51.png"/> of nonnegative degree <img class="formulaInl" alt="$ l $" src="form_94.png"/> and <code>float</code> argument <img class="formulaInl" alt="$ |x| <= 0 $" src="form_95.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>legendre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00977">977</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1b39bc22e3cc4860d08eb54099460391">◆ </a></span>legendrel()</h2>
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<td class="memname">long double std::legendrel </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
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<p>Return the Legendre polynomial <img class="formulaInl" alt="$ P_l(x) $" src="form_51.png"/> of nonnegative degree <img class="formulaInl" alt="$ l $" src="form_94.png"/> and <code>long double</code> argument <img class="formulaInl" alt="$ |x| <= 0 $" src="form_95.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>legendre for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l00987">987</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga67a6bfed9b6ab692e8c798b674431424">◆ </a></span>riemann_zeta()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::riemann_zeta </td>
<td>(</td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__s</em></td><td>)</td>
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<p>Return the Riemann zeta function <img class="formulaInl" alt="$ \zeta(s) $" src="form_98.png"/> for real argument <img class="formulaInl" alt="$ s $" src="form_99.png"/>.</p>
<p>The Riemann zeta function is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 \]" src="form_100.png"/>
</p>
<p> and </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} \hbox{ for } 0 <= s <= 1 \]" src="form_101.png"/>
</p>
<p> For s < 1 use the reflection formula: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) \]" src="form_102.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__s</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__s</td><td>The argument <code> s != 1 </code> </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01058">1058</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaf92063315061a56d3e2c4053156d968e">◆ </a></span>riemann_zetaf()</h2>
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<td class="memname">float std::riemann_zetaf </td>
<td>(</td>
<td class="paramtype">float </td>
<td class="paramname"><em>__s</em></td><td>)</td>
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<p>Return the Riemann zeta function <img class="formulaInl" alt="$ \zeta(s) $" src="form_98.png"/> for <code>float</code> argument <img class="formulaInl" alt="$ s $" src="form_99.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>riemann_zeta for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01022">1022</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1e92da3b878d75270f38d3ec9b513086">◆ </a></span>riemann_zetal()</h2>
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<td class="memname">long double std::riemann_zetal </td>
<td>(</td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__s</em></td><td>)</td>
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<p>Return the Riemann zeta function <img class="formulaInl" alt="$ \zeta(s) $" src="form_98.png"/> for <code>long double</code> argument <img class="formulaInl" alt="$ s $" src="form_99.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>riemann_zeta for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01032">1032</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga478e517ed975bcb256de230e64f0fda5">◆ </a></span>sph_bessel()</h2>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::sph_bessel </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
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<td class="paramname"><em>__x</em> </td>
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<p>Return the spherical Bessel function <img class="formulaInl" alt="$ j_n(x) $" src="form_103.png"/> of nonnegative order n and real argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The spherical Bessel function is defined by: </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) \]" src="form_104.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__n</td><td>The integral order <code> n >= 0 </code> </td></tr>
<tr><td class="paramname">__x</td><td>The real argument <code> x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01102">1102</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga534e36e1dcefad8daec98920db16eec4">◆ </a></span>sph_besself()</h2>
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<td class="memname">float std::sph_besself </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
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<td class="paramname"><em>__x</em> </td>
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<p>Return the spherical Bessel function <img class="formulaInl" alt="$ j_n(x) $" src="form_103.png"/> of nonnegative order n and <code>float</code> argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>sph_bessel for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01073">1073</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga11d72b1af81ce9da3c878a25087ee927">◆ </a></span>sph_bessell()</h2>
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<td class="memname">long double std::sph_bessell </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
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<p>Return the spherical Bessel function <img class="formulaInl" alt="$ j_n(x) $" src="form_103.png"/> of nonnegative order n and <code>long double</code> argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>sph_bessel for more details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01083">1083</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<td class="memname">__gnu_cxx::__promote<_Tp>::__type std::sph_legendre </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__m</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">_Tp </td>
<td class="paramname"><em>__theta</em> </td>
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<p>Return the spherical Legendre function of nonnegative integral degree <code>l</code> and order <code>m</code> and real angle <img class="formulaInl" alt="$ \theta $" src="form_105.png"/> in radians.</p>
<p>The spherical Legendre function is defined by </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}] P_l^m(\cos\theta) \exp^{im\phi} \]" src="form_106.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the angle <code>__theta</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__l</td><td>The order <code> __l >= 0 </code> </td></tr>
<tr><td class="paramname">__m</td><td>The degree <code> __m >= 0 </code> and <code> __m <= __l </code> </td></tr>
<tr><td class="paramname">__theta</td><td>The radian polar angle argument </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01149">1149</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#gaae635d28c06a3be2679901b382090852">◆ </a></span>sph_legendref()</h2>
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<td class="memname">float std::sph_legendref </td>
<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__m</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__theta</em> </td>
</tr>
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<td></td>
<td>)</td>
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<p>Return the spherical Legendre function of nonnegative integral degree <code>l</code> and order <code>m</code> and float angle <img class="formulaInl" alt="$ \theta $" src="form_105.png"/> in radians.</p>
<dl class="section see"><dt>See also</dt><dd>sph_legendre for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01117">1117</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga2f6618dea1847f09fd67f3c974c1910d">◆ </a></span>sph_legendrel()</h2>
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<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__l</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__m</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">long double </td>
<td class="paramname"><em>__theta</em> </td>
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<td></td>
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<p>Return the spherical Legendre function of nonnegative integral degree <code>l</code> and order <code>m</code> and <code>long double</code> angle <img class="formulaInl" alt="$ \theta $" src="form_105.png"/> in radians.</p>
<dl class="section see"><dt>See also</dt><dd>sph_legendre for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01128">1128</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga1cf4362a67ab5bae9e6b69c867e85371">◆ </a></span>sph_neumann()</h2>
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<td>(</td>
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<td class="paramname"><em>__n</em>, </td>
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<td class="paramkey"></td>
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<td class="paramname"><em>__x</em> </td>
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<p>Return the spherical Neumann function of integral order <img class="formulaInl" alt="$ n >= 0 $" src="form_107.png"/> and real argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<p>The spherical Neumann function is defined by </p><p class="formulaDsp">
<img class="formulaDsp" alt="\[ n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) \]" src="form_108.png"/>
</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_Tp</td><td>The floating-point type of the argument <code>__x</code>. </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">__n</td><td>The integral order <code> n >= 0 </code> </td></tr>
<tr><td class="paramname">__x</td><td>The real argument <code> __x >= 0 </code> </td></tr>
</table>
</dd>
</dl>
<dl class="exception"><dt>Exceptions</dt><dd>
<table class="exception">
<tr><td class="paramname">std::domain_error</td><td>if <code> __x < 0 </code>. </td></tr>
</table>
</dd>
</dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01193">1193</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga789143122fa99536329bc2d1b1aac2f0">◆ </a></span>sph_neumannf()</h2>
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<td>(</td>
<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">float </td>
<td class="paramname"><em>__x</em> </td>
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<td></td>
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<p>Return the spherical Neumann function of integral order <img class="formulaInl" alt="$ n >= 0 $" src="form_107.png"/> and <code>float</code> argument <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>sph_neumann for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01164">1164</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#ga3cededa9b6e4601f190c3811e6aabfd6">◆ </a></span>sph_neumannl()</h2>
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<td class="memname">long double std::sph_neumannl </td>
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<td class="paramtype">unsigned int </td>
<td class="paramname"><em>__n</em>, </td>
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<td class="paramtype">long double </td>
<td class="paramname"><em>__x</em> </td>
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<td>)</td>
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<p>Return the spherical Neumann function of integral order <img class="formulaInl" alt="$ n >= 0 $" src="form_107.png"/> and <code>long double</code> <img class="formulaInl" alt="$ x >= 0 $" src="form_50.png"/>.</p>
<dl class="section see"><dt>See also</dt><dd>sph_neumann for details. </dd></dl>
<p class="definition">Definition at line <a class="el" href="a00512_source.html#l01174">1174</a> of file <a class="el" href="a00512_source.html">specfun.h</a>.</p>
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