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  4. <title>Elliptic Integrals of the Third Kind - Legendre Form</title>
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  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="../zetas.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
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    25. 

  25. <div class="section" lang="en">
    26. 

  26. <div class="titlepage"><div><div><h4 class="title">
    27. 

  27. <a name="math_toolkit.special.ellint.ellint_3"></a><a class="link" href="ellint_3.html" title="Elliptic Integrals of the Third Kind - Legendre Form">Elliptic Integrals
    28. 

  28.         of the Third Kind - Legendre Form</a>
    29. 

  29. </h4></div></div></div>
    30. 

  30. <a name="math_toolkit.special.ellint.ellint_3.synopsis"></a><h6>
    31. 

  31. <a name="id1302489"></a>
    32. 

  32.           <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.synopsis">Synopsis</a>
    33. 

  33.         </h6>
    34. 

  34. <p>
    35. 

  35.           
    36. 

  36. </p>
    37. 

  37. <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
    38. 

  38. </pre>
    39. 

  39. <p>
    40. 

  40.         </p>
    41. 

  41. <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>
    42. 

  42. 

  43. <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
    44. 

  44. <a class="link" href="../../main_overview/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>
    45. 

  45. 

  46. <span class="keyword">template</span> <span class="special">&lt;</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="Policies">Policy</a><span class="special">&gt;</span>
    47. 

  47. <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&amp;);</span>
    48. 

  48. 

  49. <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
    50. 

  50. <a class="link" href="../../main_overview/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>
    51. 

  51. 

  52. <span class="keyword">template</span> <span class="special">&lt;</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="Policies">Policy</a><span class="special">&gt;</span>
    53. 

  53. <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&amp;);</span>
    54. 

  54. 

  55. <span class="special">}}</span> <span class="comment">// namespaces
    56. 

  56. </span></pre>
    57. 

  57. <a name="math_toolkit.special.ellint.ellint_3.description"></a><h6>
    58. 

  58. <a name="id1303136"></a>
    59. 

  59.           <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.description">Description</a>
    60. 

  60.         </h6>
    61. 

  61. <p>
    62. 

  62.           These two functions evaluate the incomplete elliptic integral of the third
    63. 

  63.           kind <span class="emphasis"><em>&#928;(n, &#966;, k)</em></span> and its complete counterpart <span class="emphasis"><em>&#928;(n,
    64. 

  64.           k) = E(n, &#960;/2, k)</em></span>.
    65. 

  65.         </p>
    66. 

  66. <p>
    67. 

  67.           <span class="inlinemediaobject"><img src="../../../../graphs/ellint_3.png" align="middle"></span>
    68. 

  68.         </p>
    69. 

  69. <p>
    70. 

  70.           The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
    71. 

  71.           type calculation rules</em></span></a> when the arguments are of different
    72. 

  72.           types: when they are the same type then the result is the same type as
    73. 

  73.           the arguments.
    74. 

  74.         </p>
    75. 

  75. <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
    76. 

  76. <a class="link" href="../../main_overview/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>
    77. 

  77. 

  78. <span class="keyword">template</span> <span class="special">&lt;</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="Policies">Policy</a><span class="special">&gt;</span>
    79. 

  79. <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&amp;);</span>
    80. 

  80. </pre>
    81. 

  81. <p>
    82. 

  82.           Returns the incomplete elliptic integral of the third kind <span class="emphasis"><em>&#928;(n,
    83. 

  83.           &#966;, k)</em></span>:
    84. 

  84.         </p>
    85. 

  85. <p>
    86. 

  86.           <span class="inlinemediaobject"><img src="../../../../equations/ellint4.png"></span>
    87. 

  87.         </p>
    88. 

  88. <p>
    89. 

  89.           Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span> and <span class="emphasis"><em>n &lt; 1/sin<sup>2</sup>(&#966;)</em></span>,
    90. 

  90.           otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>
    91. 

  91.           (outside this range the result would be complex).
    92. 

  92.         </p>
    93. 

  93. <p>
    94. 

  94.           The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
    95. 

  95.           optional and can be used to control the behaviour of the function: how
    96. 

  96.           it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
    97. 

  97.         </p>
    98. 

  98. <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
    99. 

  99. <a class="link" href="../../main_overview/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>
    100. 

  100. 

  101. <span class="keyword">template</span> <span class="special">&lt;</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="Policies">Policy</a><span class="special">&gt;</span>
    102. 

  102. <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&amp;);</span>
    103. 

  103. </pre>
    104. 

  104. <p>
    105. 

  105.           Returns the complete elliptic integral of the first kind <span class="emphasis"><em>&#928;(n,
    106. 

  106.           k)</em></span>:
    107. 

  107.         </p>
    108. 

  108. <p>
    109. 

  109.           <span class="inlinemediaobject"><img src="../../../../equations/ellint8.png"></span>
    110. 

  110.         </p>
    111. 

  111. <p>
    112. 

  112.           Requires <span class="emphasis"><em>-1 &lt;= k &lt;= 1</em></span> and <span class="emphasis"><em>n &lt; 1</em></span>,
    113. 

  113.           otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>
    114. 

  114.           (outside this range the result would be complex).
    115. 

  115.         </p>
    116. 

  116. <p>
    117. 

  117.           The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is
    118. 

  118.           optional and can be used to control the behaviour of the function: how
    119. 

  119.           it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
    120. 

  120.         </p>
    121. 

  121. <a name="math_toolkit.special.ellint.ellint_3.accuracy"></a><h6>
    122. 

  122. <a name="id1303730"></a>
    123. 

  123.           <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.accuracy">Accuracy</a>
    124. 

  124.         </h6>
    125. 

  125. <p>
    126. 

  126.           These functions are computed using only basic arithmetic operations, so
    127. 

  127.           there isn't much variation in accuracy over differing platforms. Note that
    128. 

  128.           only results for the widest floating point type on the system are given
    129. 

  129.           as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
    130. 

  130.           All values are relative errors in units of epsilon.
    131. 

  131.         </p>
    132. 

  132. <div class="table">
    133. 

  133. <a name="math_toolkit.special.ellint.ellint_3.errors_rates_in_the_elliptic_integrals_of_the_third_kind"></a><p class="title"><b>Table&#160;46.&#160;Errors Rates in the Elliptic Integrals of the Third Kind</b></p>
    134. 

  134. <div class="table-contents"><table class="table" summary="Errors Rates in the Elliptic Integrals of the Third Kind">
    135. 

  135. <colgroup>
    136. 

  136. <col>
    137. 

  137. <col>
    138. 

  138. <col>
    139. 

  139. <col>
    140. 

  140. </colgroup>
    141. 

  141. <thead><tr>
    142. 

  142. <th>
    143. 

  143.                   <p>
    144. 

  144.                     Significand Size
    145. 

  145.                   </p>
    146. 

  146.                 </th>
    147. 

  147. <th>
    148. 

  148.                   <p>
    149. 

  149.                     Platform and Compiler
    150. 

  150.                   </p>
    151. 

  151.                 </th>
    152. 

  152. <th>
    153. 

  153.                   <p>
    154. 

  154.                     &#928;(n, &#966;, k)
    155. 

  155.                   </p>
    156. 

  156.                 </th>
    157. 

  157. <th>
    158. 

  158.                   <p>
    159. 

  159.                     &#928;(n, k)
    160. 

  160.                   </p>
    161. 

  161.                 </th>
    162. 

  162. </tr></thead>
    163. 

  163. <tbody>
    164. 

  164. <tr>
    165. 

  165. <td>
    166. 

  166.                   <p>
    167. 

  167.                     53
    168. 

  168.                   </p>
    169. 

  169.                 </td>
    170. 

  170. <td>
    171. 

  171.                   <p>
    172. 

  172.                     Win32 / Visual C++ 8.0
    173. 

  173.                   </p>
    174. 

  174.                 </td>
    175. 

  175. <td>
    176. 

  176.                   <p>
    177. 

  177.                     Peak=29 Mean=2.2
    178. 

  178.                   </p>
    179. 

  179.                 </td>
    180. 

  180. <td>
    181. 

  181.                   <p>
    182. 

  182.                     Peak=3 Mean=0.8
    183. 

  183.                   </p>
    184. 

  184.                 </td>
    185. 

  185. </tr>
    186. 

  186. <tr>
    187. 

  187. <td>
    188. 

  188.                   <p>
    189. 

  189.                     64
    190. 

  190.                   </p>
    191. 

  191.                 </td>
    192. 

  192. <td>
    193. 

  193.                   <p>
    194. 

  194.                     Red Hat Linux / G++ 3.4
    195. 

  195.                   </p>
    196. 

  196.                 </td>
    197. 

  197. <td>
    198. 

  198.                   <p>
    199. 

  199.                     Peak=14 Mean=1.3
    200. 

  200.                   </p>
    201. 

  201.                 </td>
    202. 

  202. <td>
    203. 

  203.                   <p>
    204. 

  204.                     Peak=2.3 Mean=0.8
    205. 

  205.                   </p>
    206. 

  206.                 </td>
    207. 

  207. </tr>
    208. 

  208. <tr>
    209. 

  209. <td>
    210. 

  210.                   <p>
    211. 

  211.                     113
    212. 

  212.                   </p>
    213. 

  213.                 </td>
    214. 

  214. <td>
    215. 

  215.                   <p>
    216. 

  216.                     HP-UX / HP aCC 6
    217. 

  217.                   </p>
    218. 

  218.                 </td>
    219. 

  219. <td>
    220. 

  220.                   <p>
    221. 

  221.                     Peak=10 Mean=1.4
    222. 

  222.                   </p>
    223. 

  223.                 </td>
    224. 

  224. <td>
    225. 

  225.                   <p>
    226. 

  226.                     Peak=4.2 Mean=1.1
    227. 

  227.                   </p>
    228. 

  228.                 </td>
    229. 

  229. </tr>
    230. 

  230. </tbody>
    231. 

  231. </table></div>
    232. 

  232. </div>
    233. 

  233. <br class="table-break"><a name="math_toolkit.special.ellint.ellint_3.testing"></a><h6>
    234. 

  234. <a name="id1303904"></a>
    235. 

  235.           <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.testing">Testing</a>
    236. 

  236.         </h6>
    237. 

  237. <p>
    238. 

  238.           The tests use a mixture of spot test values calculated using the online
    239. 

  239.           calculator at <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
    240. 

  240.           and random test data generated using NTL::RR at 1000-bit precision and
    241. 

  241.           this implementation.
    242. 

  242.         </p>
    243. 

  243. <a name="math_toolkit.special.ellint.ellint_3.implementation"></a><h6>
    244. 

  244. <a name="id1303926"></a>
    245. 

  245.           <a class="link" href="ellint_3.html#math_toolkit.special.ellint.ellint_3.implementation">Implementation</a>
    246. 

  246.         </h6>
    247. 

  247. <p>
    248. 

  248.           The implementation for &#928;(n, &#966;, k) first siphons off the special cases:
    249. 

  249.         </p>
    250. 

  250. <p>
    251. 

  251.           <span class="emphasis"><em>&#928;(0, &#966;, k) = F(&#966;, k)</em></span>
    252. 

  252.         </p>
    253. 

  253. <p>
    254. 

  254.           <span class="emphasis"><em>&#928;(n, &#960;/2, k) = &#928;(n, k)</em></span>
    255. 

  255.         </p>
    256. 

  256. <p>
    257. 

  257.           and
    258. 

  258.         </p>
    259. 

  259. <p>
    260. 

  260.           <span class="inlinemediaobject"><img src="../../../../equations/ellint23.png"></span>
    261. 

  261.         </p>
    262. 

  262. <p>
    263. 

  263.           Then if n &lt; 0 the relations (A&amp;S 17.7.15/16):
    264. 

  264.         </p>
    265. 

  265. <p>
    266. 

  266.           <span class="inlinemediaobject"><img src="../../../../equations/ellint24.png"></span>
    267. 

  267.         </p>
    268. 

  268. <p>
    269. 

  269.           are used to shift <span class="emphasis"><em>n</em></span> to the range [0, 1].
    270. 

  270.         </p>
    271. 

  271. <p>
    272. 

  272.           Then the relations:
    273. 

  273.         </p>
    274. 

  274. <p>
    275. 

  275.           <span class="emphasis"><em>&#928;(n, -&#966;, k) = -&#928;(n, &#966;, k)</em></span>
    276. 

  276.         </p>
    277. 

  277. <p>
    278. 

  278.           <span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) + 2m&#928;(n, k) ; n &lt;= 1</em></span>
    279. 

  279.         </p>
    280. 

  280. <p>
    281. 

  281.           <span class="emphasis"><em>&#928;(n, &#966;+m&#960;, k) = &#928;(n, &#966;, k) ; n &gt; 1</em></span> <sup>[<a name="id1304041" href="#ftn.id1304041" class="footnote">1</a>]</sup>
    282. 

  282.         </p>
    283. 

  283. <p>
    284. 

  284.           are used to move &#966; &#160; to the range [0, &#960;/2].
    285. 

  285.         </p>
    286. 

  286. <p>
    287. 

  287.           The functions are then implemented in terms of Carlson's integrals using
    288. 

  288.           the relations:
    289. 

  289.         </p>
    290. 

  290. <p>
    291. 

  291.           <span class="inlinemediaobject"><img src="../../../../equations/ellint25.png"></span>
    292. 

  292.         </p>
    293. 

  293. <p>
    294. 

  294.           and
    295. 

  295.         </p>
    296. 

  296. <p>
    297. 

  297.           <span class="inlinemediaobject"><img src="../../../../equations/ellint26.png"></span>
    298. 

  298.         </p>
    299. 

  299. <div class="footnotes">
    300. 

  300. <br><hr width="100" align="left">
    301. 

  301. <div class="footnote"><p><sup>[<a name="ftn.id1304041" href="#id1304041" class="para">1</a>] </sup>
    302. 

  302.             I haven't been able to find a literature reference for this relation,
    303. 

  303.             but it appears to be the convention used by Mathematica. Intuitively
    304. 

  304.             the first <span class="emphasis"><em>2 * m * &#928;(n, k)</em></span> terms cancel out as the
    305. 

  305.             derivative alternates between +&#8734; and -&#8734;.
    306. 

  306.           </p></div>
    307. 

  307. </div>
    308. 

  308. </div>
    309. 

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    310. 

  310. <td align="left"></td>
    311. 

  311. <td align="right"><div class="copyright-footer">Copyright &#169; 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
    312. 

  312.       Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan R&#229;de, Gautam Sewani and
    313. 

  313.       Thijs van den Berg<p>
    314. 

  314.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    315. 

  315.         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>)
    316. 

  316.       </p>
    317. 

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    318. 

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    319. 

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