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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Elliptic Integrals of the Third Kind - Legendre Form</title> | |
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| 13 | <table cellpadding="2" width="100%"><tr> | |
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| 20 | </tr></table> | |
| 21 | <hr> | |
| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="ellint_2.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ellint_d.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h3 class="title"> | |
| 27 | <a name="math_toolkit.ellint.ellint_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals of the | |
| 28 | Third Kind - Legendre Form</a> | |
| 29 | </h3></div></div></div> | |
| 30 | <h5> | |
| 31 | <a name="math_toolkit.ellint.ellint_3.h0"></a> | |
| 32 | <span class="phrase"><a name="math_toolkit.ellint.ellint_3.synopsis"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.synopsis">Synopsis</a> | |
| 33 | </h5> | |
| 34 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ellint_3</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 35 | </pre> | |
| 36 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> | |
| 37 | ||
| 38 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
| 39 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span> | |
| 40 | ||
| 41 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 42 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 43 | ||
| 44 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> | |
| 45 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span> | |
| 46 | ||
| 47 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 48 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 49 | ||
| 50 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
| 51 | </pre> | |
| 52 | <h5> | |
| 53 | <a name="math_toolkit.ellint.ellint_3.h1"></a> | |
| 54 | <span class="phrase"><a name="math_toolkit.ellint.ellint_3.description"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.description">Description</a> | |
| 55 | </h5> | |
| 56 | <p> | |
| 57 | These two functions evaluate the incomplete elliptic integral of the third | |
| 58 | kind <span class="emphasis"><em>Π(n, φ, k)</em></span> and its complete counterpart <span class="emphasis"><em>Π(n, | |
| 59 | k) = E(n, π/2, k)</em></span>. | |
| 60 | </p> | |
| 61 | <p> | |
| 62 | <span class="inlinemediaobject"><img src="../../../graphs/ellint_3.svg" align="middle"></span> | |
| 63 | </p> | |
| 64 | <p> | |
| 65 | The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result | |
| 66 | type calculation rules</em></span></a> when the arguments are of different | |
| 67 | types: when they are the same type then the result is the same type as the | |
| 68 | arguments. | |
| 69 | </p> | |
| 70 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> | |
| 71 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">);</span> | |
| 72 | ||
| 73 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 74 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">phi</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 75 | </pre> | |
| 76 | <p> | |
| 77 | Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>Π(n, | |
| 78 | φ, k)</em></span>: | |
| 79 | </p> | |
| 80 | <p> | |
| 81 | <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span> | |
| 82 | </p> | |
| 83 | <p> | |
| 84 | Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> and <span class="emphasis"><em>n < 1/sin<sup>2</sup>(φ)</em></span>, | |
| 85 | otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
| 86 | (outside this range the result would be complex). | |
| 87 | </p> | |
| 88 | <p> | |
| 89 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 90 | be used to control the behaviour of the function: how it handles errors, | |
| 91 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 92 | documentation for more details</a>. | |
| 93 | </p> | |
| 94 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span> | |
| 95 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">);</span> | |
| 96 | ||
| 97 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 98 | <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ellint_3</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 99 | </pre> | |
| 100 | <p> | |
| 101 | Returns the complete elliptic integral of the first kind <span class="emphasis"><em>Π(n, k)</em></span>: | |
| 102 | </p> | |
| 103 | <p> | |
| 104 | <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span> | |
| 105 | </p> | |
| 106 | <p> | |
| 107 | Requires <span class="emphasis"><em>-1 <= k <= 1</em></span> and <span class="emphasis"><em>n < 1</em></span>, | |
| 108 | otherwise returns the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
| 109 | (outside this range the result would be complex). | |
| 110 | </p> | |
| 111 | <p> | |
| 112 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 113 | be used to control the behaviour of the function: how it handles errors, | |
| 114 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 115 | documentation for more details</a>. | |
| 116 | </p> | |
| 117 | <h5> | |
| 118 | <a name="math_toolkit.ellint.ellint_3.h2"></a> | |
| 119 | <span class="phrase"><a name="math_toolkit.ellint.ellint_3.accuracy"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.accuracy">Accuracy</a> | |
| 120 | </h5> | |
| 121 | <p> | |
| 122 | These functions are computed using only basic arithmetic operations, so there | |
| 123 | isn't much variation in accuracy over differing platforms. Note that only | |
| 124 | results for the widest floating point type on the system are given as narrower | |
| 125 | types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively | |
| 126 | zero error</a>. All values are relative errors in units of epsilon. | |
| 127 | </p> | |
| 128 | <div class="table"> | |
| 129 | <a name="math_toolkit.ellint.ellint_3.table_ellint_3"></a><p class="title"><b>Table 6.65. Error rates for ellint_3</b></p> | |
| 130 | <div class="table-contents"><table class="table" summary="Error rates for ellint_3"> | |
| 131 | <colgroup> | |
| 132 | <col> | |
| 133 | <col> | |
| 134 | <col> | |
| 135 | <col> | |
| 136 | <col> | |
| 137 | </colgroup> | |
| 138 | <thead><tr> | |
| 139 | <th> | |
| 140 | </th> | |
| 141 | <th> | |
| 142 | <p> | |
| 143 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 144 | </p> | |
| 145 | </th> | |
| 146 | <th> | |
| 147 | <p> | |
| 148 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 149 | </p> | |
| 150 | </th> | |
| 151 | <th> | |
| 152 | <p> | |
| 153 | GNU C++ version 5.1.0<br> linux<br> double | |
| 154 | </p> | |
| 155 | </th> | |
| 156 | <th> | |
| 157 | <p> | |
| 158 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 159 | </p> | |
| 160 | </th> | |
| 161 | </tr></thead> | |
| 162 | <tbody> | |
| 163 | <tr> | |
| 164 | <td> | |
| 165 | <p> | |
| 166 | Elliptic Integral PI: Mathworld Data | |
| 167 | </p> | |
| 168 | </td> | |
| 169 | <td> | |
| 170 | <p> | |
| 171 | <span class="blue">Max = 565ε (Mean = 102ε)</span> | |
| 172 | </p> | |
| 173 | </td> | |
| 174 | <td> | |
| 175 | <p> | |
| 176 | <span class="blue">Max = 475ε (Mean = 86.3ε)</span><br> <br> | |
| 177 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
| 178 | = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_ellint_3__tr1_cmath__Elliptic_Integral_PI_Mathworld_Data">And | |
| 179 | other failures.</a>)</span> | |
| 180 | </p> | |
| 181 | </td> | |
| 182 | <td> | |
| 183 | <p> | |
| 184 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
| 185 | 1.16:</em></span> Max = 1.48e+05ε (Mean = 2.54e+04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ellint_3_GSL_1_16_Elliptic_Integral_PI_Mathworld_Data">And | |
| 186 | other failures.</a>) | |
| 187 | </p> | |
| 188 | </td> | |
| 189 | <td> | |
| 190 | <p> | |
| 191 | <span class="blue">Max = 475ε (Mean = 86.3ε)</span> | |
| 192 | </p> | |
| 193 | </td> | |
| 194 | </tr> | |
| 195 | <tr> | |
| 196 | <td> | |
| 197 | <p> | |
| 198 | Elliptic Integral PI: Random Data | |
| 199 | </p> | |
| 200 | </td> | |
| 201 | <td> | |
| 202 | <p> | |
| 203 | <span class="blue">Max = 9.08ε (Mean = 0.99ε)</span> | |
| 204 | </p> | |
| 205 | </td> | |
| 206 | <td> | |
| 207 | <p> | |
| 208 | <span class="blue">Max = 4.54ε (Mean = 0.895ε)</span><br> <br> | |
| 209 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
| 210 | = 3.37e+20ε (Mean = 3.47e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_ellint_3__tr1_cmath__Elliptic_Integral_PI_Random_Data">And | |
| 211 | other failures.</a>)</span> | |
| 212 | </p> | |
| 213 | </td> | |
| 214 | <td> | |
| 215 | <p> | |
| 216 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL | |
| 217 | 1.16:</em></span> Max = 633ε (Mean = 50.1ε)) | |
| 218 | </p> | |
| 219 | </td> | |
| 220 | <td> | |
| 221 | <p> | |
| 222 | <span class="blue">Max = 4.49ε (Mean = 0.891ε)</span> | |
| 223 | </p> | |
| 224 | </td> | |
| 225 | </tr> | |
| 226 | <tr> | |
| 227 | <td> | |
| 228 | <p> | |
| 229 | Elliptic Integral PI: Large Random Data | |
| 230 | </p> | |
| 231 | </td> | |
| 232 | <td> | |
| 233 | <p> | |
| 234 | <span class="blue">Max = 2.86ε (Mean = 0.944ε)</span> | |
| 235 | </p> | |
| 236 | </td> | |
| 237 | <td> | |
| 238 | <p> | |
| 239 | <span class="blue">Max = 3.7ε (Mean = 0.893ε)</span><br> <br> | |
| 240 | (<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max | |
| 241 | = 2.52e+18ε (Mean = 4.83e+17ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_ellint_3__tr1_cmath__Elliptic_Integral_PI_Large_Random_Data">And | |
| 242 | other failures.</a>)</span> | |
| 243 | </p> | |
| 244 | </td> | |
| 245 | <td> | |
| 246 | <p> | |
| 247 | <span class="blue">Max = 0.557ε (Mean = 0.0389ε)</span><br> | |
| 248 | <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 40.1ε (Mean = 7.77ε)) | |
| 249 | </p> | |
| 250 | </td> | |
| 251 | <td> | |
| 252 | <p> | |
| 253 | <span class="blue">Max = 3.7ε (Mean = 0.892ε)</span> | |
| 254 | </p> | |
| 255 | </td> | |
| 256 | </tr> | |
| 257 | </tbody> | |
| 258 | </table></div> | |
| 259 | </div> | |
| 260 | <br class="table-break"><h5> | |
| 261 | <a name="math_toolkit.ellint.ellint_3.h3"></a> | |
| 262 | <span class="phrase"><a name="math_toolkit.ellint.ellint_3.testing"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.testing">Testing</a> | |
| 263 | </h5> | |
| 264 | <p> | |
| 265 | The tests use a mixture of spot test values calculated using the online calculator | |
| 266 | at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, | |
| 267 | and random test data generated using NTL::RR at 1000-bit precision and this | |
| 268 | implementation. | |
| 269 | </p> | |
| 270 | <h5> | |
| 271 | <a name="math_toolkit.ellint.ellint_3.h4"></a> | |
| 272 | <span class="phrase"><a name="math_toolkit.ellint.ellint_3.implementation"></a></span><a class="link" href="ellint_3.html#math_toolkit.ellint.ellint_3.implementation">Implementation</a> | |
| 273 | </h5> | |
| 274 | <p> | |
| 275 | The implementation for Π(n, φ, k) first siphons off the special cases: | |
| 276 | </p> | |
| 277 | <p> | |
| 278 | <span class="emphasis"><em>Π(0, φ, k) = F(φ, k)</em></span> | |
| 279 | </p> | |
| 280 | <p> | |
| 281 | <span class="emphasis"><em>Π(n, π/2, k) = Π(n, k)</em></span> | |
| 282 | </p> | |
| 283 | <p> | |
| 284 | and | |
| 285 | </p> | |
| 286 | <p> | |
| 287 | <span class="inlinemediaobject"><img src="../../../equations/ellint23.svg"></span> | |
| 288 | </p> | |
| 289 | <p> | |
| 290 | Then if n < 0 the relations (A&S 17.7.15/16): | |
| 291 | </p> | |
| 292 | <p> | |
| 293 | <span class="inlinemediaobject"><img src="../../../equations/ellint24.svg"></span> | |
| 294 | </p> | |
| 295 | <p> | |
| 296 | are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1]. | |
| 297 | </p> | |
| 298 | <p> | |
| 299 | Then the relations: | |
| 300 | </p> | |
| 301 | <p> | |
| 302 | <span class="emphasis"><em>Π(n, -φ, k) = -Π(n, φ, k)</em></span> | |
| 303 | </p> | |
| 304 | <p> | |
| 305 | <span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) + 2mΠ(n, k) ; n <= 1</em></span> | |
| 306 | </p> | |
| 307 | <p> | |
| 308 | <span class="emphasis"><em>Π(n, φ+mπ, k) = Π(n, φ, k) ; n > 1</em></span> <a href="#ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><sup class="footnote"><a name="math_toolkit.ellint.ellint_3.f0"></a>[1]</sup></a> | |
| 309 | </p> | |
| 310 | <p> | |
| 311 | are used to move φ   to the range [0, π/2]. | |
| 312 | </p> | |
| 313 | <p> | |
| 314 | The functions are then implemented in terms of Carlson's integrals using | |
| 315 | the relations: | |
| 316 | </p> | |
| 317 | <p> | |
| 318 | <span class="inlinemediaobject"><img src="../../../equations/ellint25.svg"></span> | |
| 319 | </p> | |
| 320 | <p> | |
| 321 | and | |
| 322 | </p> | |
| 323 | <p> | |
| 324 | <span class="inlinemediaobject"><img src="../../../equations/ellint26.svg"></span> | |
| 325 | </p> | |
| 326 | <div class="footnotes"> | |
| 327 | <br><hr style="width:100; align:left;"> | |
| 328 | <div id="ftn.math_toolkit.ellint.ellint_3.f0" class="footnote"><p><a href="#math_toolkit.ellint.ellint_3.f0" class="para"><sup class="para">[1] </sup></a> | |
| 329 | I haven't been able to find a literature reference for this relation, but | |
| 330 | it appears to be the convention used by Mathematica. Intuitively the first | |
| 331 | <span class="emphasis"><em>2 * m * Π(n, k)</em></span> terms cancel out as the derivative | |
| 332 | alternates between +∞ and -∞. | |
| 333 | </p></div> | |
| 334 | </div> | |
| 335 | </div> | |
| 336 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 337 | <td align="left"></td> | |
| 338 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 339 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 340 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 341 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 342 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 343 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
| 344 | </p> | |
| 345 | </div></td> | |
| 346 | </tr></table> | |
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