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<title>std::cyl_bessel_k, std::cyl_bessel_kf, std::cyl_bessel_kl</title>
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>cyl_bessel_k, <span style="font-size:0.7em; line-height:130%">std::</span>cyl_bessel_kf, <span style="font-size:0.7em; line-height:130%">std::</span>cyl_bessel_kl</h1>
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<div id="contentSub"><span class="subpages">&lt; <a href="cpp.html">cpp</a>&lrm; | <a href="cpp-numeric.html">numeric</a>&lrm; | <a href="cpp-numeric-special_functions.html">special functions</a></span></div>
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<div>&#23450;&#20041;&#20110;&#22836;&#25991;&#20214; <code><a href="cpp-header-cmath.html">&lt;cmath&gt;</a></code></div>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> &nbsp; &nbsp; &nbsp;cyl_bessel_k<span class="br0">(</span> <span class="kw4">double</span> n, <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span><br /></span>
<p><span class="kw4">float</span> &nbsp; &nbsp; &nbsp; cyl_bessel_kf<span class="br0">(</span> <span class="kw4">float</span> n, <span class="kw4">float</span> x &nbsp;<span class="br0">)</span><span class="sy4">;</span><br /></p>
<span class="kw4">long</span> <span class="kw4">double</span> cyl_bessel_kl<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> n, <span class="kw4">long</span> <span class="kw4">double</span> x <span class="br0">)</span><span class="sy4">;</span></div>
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<td>(1)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div><span class="mw-geshi cpp source-cpp">Promoted &nbsp; &nbsp;cyl_bessel_k<span class="br0">(</span> Arithmetic n, Arithmetic x <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(2)</td>
<td><span class="t-mark-rev t-since-cxx17">(C++17 &#36215;)</span></td>
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<div class="t-li1"><span class="t-li">1)</span> &#35745;&#31639; <code>n</code> &#21644; <code>x</code> &#30340;<a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1_.2C_K.CE.B1" class="extiw" title="enwiki:Bessel function">&#38750;&#24120;&#35268;&#20462;&#27491;&#26609;&#36125;&#22622;&#23572;&#20989;&#25968;</a>&#65288;&#20134;&#31216;&#20026;&#31532;&#20108;&#31867;&#20462;&#27491;&#36125;&#22622;&#23572;&#20989;&#25968;&#65289;&#12290;</div>
<div class="t-li1"><span class="t-li">2)</span> &#23545;&#20110;&#25152;&#26377; <span class="t-v">(1)</span> &#25152;&#19981;&#35206;&#30422;&#30340;&#31639;&#26415;&#31867;&#22411;&#21442;&#25968;&#32452;&#21512;&#30340;&#37325;&#36733;&#38598;&#25110;&#20989;&#25968;&#27169;&#26495;&#12290;&#33509;&#20219;&#20309;&#21442;&#25968;&#25317;&#26377;<a href="cpp-types-is_integral.html">&#25972;&#25968;&#31867;&#22411;</a>&#65292;&#21017;&#23427;&#34987;&#36716;&#22411;&#21040; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> &#12290;&#33509;&#20219;&#20309;&#21442;&#25968;&#20026; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> &#65292;&#21017;&#36820;&#22238;&#31867;&#22411; <code>Promoted</code> &#20134;&#20026; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> &#65292;&#21542;&#21017;&#36820;&#22238;&#31867;&#22411;&#22987;&#32456;&#26159; <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> &#12290;</div>
<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">&#21442;&#25968;</span></h3>
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<td>n</td>
<td>-</td>
<td>&#20989;&#25968;&#30340;&#38454;&#25968;</td>
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<td>x</td>
<td>-</td>
<td>&#20989;&#25968;&#30340;&#21442;&#25968;</td>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">&#36820;&#22238;&#20540;</span></h3>
&#33509;&#26080;&#38169;&#35823;&#21457;&#29983;&#65292;&#21017;&#36820;&#22238; <code>n</code> &#19982; <code>x</code> &#30340;&#38750;&#24120;&#35268;&#20462;&#27491;&#26609;&#36125;&#22622;&#23572;&#20989;&#25968;&#65288;&#31532;&#20108;&#31867;&#20462;&#27491;&#36125;&#22622;&#23572;&#20989;&#25968;&#65289;&#30340;&#20540;&#65292;&#21363;&#23545;&#20110; <span class="texhtml" style="white-space: nowrap;">x&ge;0</span> &#21450;&#38750;&#25972;&#25968; n &#30340; <span class="texhtml" style="white-space: nowrap;">K<span class="t-su t-su-b"><br />
n</span>(x) = <span class="t-mfrac"></span></span>
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<td>&pi;</td>
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<td>2</td>
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<span class="t-mfrac"></span>
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<td>I<span class="t-su t-su-b"><br />
-n</span>(x)-I<span class="t-su t-su-b"><br />
n</span>(x)</td>
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<td>sin(n&pi;)</td>
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&#65288;&#20854;&#20013; <span class="texhtml" style="white-space: nowrap;">I<span class="t-su t-su-b"><br />
n</span>(x)</span> &#20026; <span class="t-lc">std::cyl_bessel_i(n,x)</span>&#65289;&#65307;&#23545;&#25972;&#25968; n &#20351;&#29992;&#26497;&#38480;&#12290;
<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">&#38169;&#35823;&#22788;&#29702;</span></h3>
<p>&#21487;&#33021;&#25253;&#21578; <a href="cpp-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> &#20013;&#25351;&#23450;&#30340;&#38169;&#35823;</p>
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<li>&#33509;&#21442;&#25968;&#26159; NaN &#65292;&#21017;&#36820;&#22238; NaN &#19988;&#19981;&#25253;&#21578;&#23450;&#20041;&#22495;&#38169;&#35823;</li>
<li>&#33509; <span class="texhtml" style="white-space: nowrap;">n&gt;=128</span> &#65292;&#21017;&#34892;&#20026;&#26159;&#23454;&#29616;&#23450;&#20041;&#30340;</li>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">&#27880;&#35299;</span></h3>
<p>&#19981;&#25903;&#25345; C++17 &#65292;&#20294;&#25903;&#25345; <a href="cpp-experimental-special_functions.html">ISO 29124:2010</a> &#30340;&#23454;&#29616;&#20250;&#25552;&#20379;&#27492;&#20989;&#25968;&#65292;&#33509;&#23454;&#29616;&#23450;&#20041;&#20102; <code>__STDCPP_MATH_SPEC_FUNCS__</code> &#20026;&#33267;&#23569; 201003L &#30340;&#20540;&#65292;&#19988;&#29992;&#25143;&#22312;&#21253;&#21547;&#20219;&#20309;&#26631;&#20934;&#24211;&#22836;&#25991;&#20214;&#21069;&#23450;&#20041;&#20102; <code>__STDCPP_WANT_MATH_SPEC_FUNCS__</code> &#12290;</p>
<p>&#19981;&#25903;&#25345; ISO 29124:2010 &#20294;&#25903;&#25345; TR 19768:2007 (TR1) &#30340;&#23454;&#29616;&#65292;&#22312;&#22836;&#25991;&#20214; <code>tr1/cmath</code> &#21450;&#21629;&#21517;&#31354;&#38388; <code>std::tr1</code> &#20013;&#25552;&#20379;&#27492;&#20989;&#25968;&#12290;</p>
<p>&#27492;&#20989;&#25968;&#30340;&#19968;&#31181;&#23454;&#29616;&#20134;<a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/1_62_0/libs/math/doc/html/math_toolkit/bessel/mbessel.html">&#21487;&#29992;&#20110; boost.math</a></p>
<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">&#31034;&#20363;</span></h3>
<div class="t-example">
<div class="t-example-live-link"></div>
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<div class="cpp source-cpp">
<pre class="de1"><span class="co2">#include &lt;cmath&gt;</span>
<span class="co2">#include &lt;iostream&gt;</span>
<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
<span class="br0">{</span>
<span class="kw4">double</span> pi <span class="sy1">=</span> <a href="cpp-numeric-math-acos.html"><span class="kw1337">std::<span class="me2">acos</span></span></a><span class="br0">(</span><span class="sy2">-</span><span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
<span class="kw4">double</span> x <span class="sy1">=</span> <span class="nu16">1.2345</span><span class="sy4">;</span>
<span class="co1">// &#23545; n == 0.5 &#28857;&#26816;&#26597;</span>
<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1">&lt;&lt;</span> <span class="st0">&quot;K_.5(&quot;</span> <span class="sy1">&lt;&lt;</span> x <span class="sy1">&lt;&lt;</span> <span class="st0">&quot;) = &quot;</span> <span class="sy1">&lt;&lt;</span> <a href="cpp-numeric-special_math-cyl_bessel_k.html"><span class="kw1411">std::<span class="me2">cyl_bessel_k</span></span></a><span class="br0">(</span> <span class="nu18">.5</span>, x<span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span>
<span class="sy1">&lt;&lt;</span> <span class="st0">&quot;calculated via I = &quot;</span> <span class="sy1">&lt;&lt;</span>
<span class="br0">(</span>pi<span class="sy2">/</span><span class="nu0">2</span><span class="br0">)</span><span class="sy2">*</span><span class="br0">(</span><a href="cpp-numeric-special_math-cyl_bessel_i.html"><span class="kw1405">std::<span class="me2">cyl_bessel_i</span></span></a><span class="br0">(</span><span class="sy2">-</span><span class="nu18">.5</span>,x<span class="br0">)</span>
<span class="sy2">-</span><a href="cpp-numeric-special_math-cyl_bessel_i.html"><span class="kw1405">std::<span class="me2">cyl_bessel_i</span></span></a><span class="br0">(</span><span class="nu18">.5</span>,x<span class="br0">)</span><span class="br0">)</span><span class="sy2">/</span><a href="cpp-numeric-math-sin.html"><span class="kw1333">std::<span class="me2">sin</span></span></a><span class="br0">(</span><span class="nu18">.5</span><span class="sy2">*</span>pi<span class="br0">)</span> <span class="sy1">&lt;&lt;</span> <span class="st0">&#39;<span class="es1">\n</span>&#39;</span><span class="sy4">;</span>
<span class="br0">}</span></pre></div>
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<p>&#36755;&#20986;&#65306;</p>
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<div class="text source-text">
<pre class="de1">K_.5(1.2345) = 0.32823
calculated via I = 0.32823</pre></div>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">&#22806;&#37096;&#38142;&#25509;</span></h3>
<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html">Weisstein, Eric W. &ldquo;&#31532;&#20108;&#31867;&#20462;&#27491;&#36125;&#22622;&#23572;&#20989;&#25968;&rdquo;</a>&#26469;&#33258; MathWorld--A Wolfram Web Resource &#12290;</p>
<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">&#21442;&#38405;</span></h3>
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<div class="t-dsc-member-div">
<div><a href="cpp-numeric-special_functions-cyl_bessel_i.html"><span class="t-lines"><span>cyl_bessel_i</span><span>cyl_bessel_if</span><span>cyl_bessel_il</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span></span></div>
</div>
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<td>&#35268;&#21017;&#21464;&#24418;&#22278;&#26609;&#36125;&#22622;&#23572;&#20989;&#25968;<br />
<span class="t-mark">(&#20989;&#25968;)</span></td>
</tr>
<tr class="t-dsc">
<td>
<div class="t-dsc-member-div">
<div><a href="cpp-numeric-special_functions-cyl_bessel_j.html"><span class="t-lines"><span>cyl_bessel_j</span><span>cyl_bessel_jf</span><span>cyl_bessel_jl</span></span></a></div>
<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span><span><span class="t-mark-rev t-since-cxx17">(C++17)</span></span></span></div>
</div>
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<td>&#65288;&#31532;&#19968;&#31867;&#65289;&#22278;&#26609;&#36125;&#22622;&#23572;&#20989;&#25968;<br />
<span class="t-mark">(&#20989;&#25968;)</span></td>
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<div class="visualClear"></div>
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