1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Derivatives of the Bessel Functions</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../bessel.html" title="Bessel Functions"> 9<link rel="prev" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds"> 10<link rel="next" href="../hankel.html" title="Hankel Functions"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="sph_bessel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="../hankel.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.bessel.bessel_derivatives"></a><a class="link" href="bessel_derivatives.html" title="Derivatives of the Bessel Functions">Derivatives of 28 the Bessel Functions</a> 29</h3></div></div></div> 30<h5> 31<a name="math_toolkit.bessel.bessel_derivatives.h0"></a> 32 <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.synopsis"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.synopsis">Synopsis</a> 33 </h5> 34<p> 35 <code class="computeroutput"><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">bessel_prime</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code> 36 </p> 37<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> 38<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">cyl_bessel_j_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 39 40<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<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">cyl_bessel_j_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 42 43<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> 44<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">cyl_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 45 46<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 47<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">cyl_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 48 49<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> 50<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">cyl_bessel_i_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 51 52<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 53<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">cyl_bessel_i_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 54 55<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> 56<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">cyl_bessel_k_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 57 58<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 59<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">cyl_bessel_k_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 60 61<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> 62<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">sph_bessel_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 63 64<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 65<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">sph_bessel_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 66 67<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> 68<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">sph_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> 69 70<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><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">sph_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 72</pre> 73<h5> 74<a name="math_toolkit.bessel.bessel_derivatives.h1"></a> 75 <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.description"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.description">Description</a> 76 </h5> 77<p> 78 These functions return the first derivative with respect to <span class="emphasis"><em>x</em></span> 79 of the corresponding Bessel function. 80 </p> 81<p> 82 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 83 type calculation rules</em></span></a> when T1 and T2 are different types. 84 The functions are also optimised for the relatively common case that T1 is 85 an integer. 86 </p> 87<p> 88 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 89 be used to control the behaviour of the function: how it handles errors, 90 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 91 documentation for more details</a>. 92 </p> 93<p> 94 The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 95 whenever the result is undefined or complex. 96 </p> 97<h5> 98<a name="math_toolkit.bessel.bessel_derivatives.h2"></a> 99 <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.testing"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.testing">Testing</a> 100 </h5> 101<p> 102 There are two sets of test values: spot values calculated using <a href="http://www.wolframalpha.com/" target="_top">wolframalpha.com</a>, 103 and a much larger set of tests computed using a relation to the underlying 104 Bessel functions that the implementation does not use. 105 </p> 106<h5> 107<a name="math_toolkit.bessel.bessel_derivatives.h3"></a> 108 <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.accuracy"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.accuracy">Accuracy</a> 109 </h5> 110<p> 111 The accuracy of these functions is broadly similar to the underlying Bessel 112 functions. 113 </p> 114<div class="table"> 115<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_"></a><p class="title"><b>Table 8.50. Error rates for cyl_bessel_i_prime (integer orders)</b></p> 116<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i_prime (integer orders)"> 117<colgroup> 118<col> 119<col> 120<col> 121<col> 122<col> 123</colgroup> 124<thead><tr> 125<th> 126 </th> 127<th> 128 <p> 129 GNU C++ version 7.1.0<br> linux<br> double 130 </p> 131 </th> 132<th> 133 <p> 134 GNU C++ version 7.1.0<br> linux<br> long double 135 </p> 136 </th> 137<th> 138 <p> 139 Sun compiler version 0x5150<br> Sun Solaris<br> long double 140 </p> 141 </th> 142<th> 143 <p> 144 Microsoft Visual C++ version 14.1<br> Win32<br> double 145 </p> 146 </th> 147</tr></thead> 148<tbody> 149<tr> 150<td> 151 <p> 152 Bessel I'0: Mathworld Data (Integer Version) 153 </p> 154 </td> 155<td> 156 <p> 157 <span class="blue">Max = 0ε (Mean = 0ε)</span> 158 </p> 159 </td> 160<td> 161 <p> 162 <span class="blue">Max = 0.82ε (Mean = 0.259ε)</span> 163 </p> 164 </td> 165<td> 166 <p> 167 <span class="blue">Max = 0.82ε (Mean = 0.259ε)</span> 168 </p> 169 </td> 170<td> 171 <p> 172 <span class="blue">Max = 0.82ε (Mean = 0.354ε)</span> 173 </p> 174 </td> 175</tr> 176<tr> 177<td> 178 <p> 179 Bessel I'1: Mathworld Data (Integer Version) 180 </p> 181 </td> 182<td> 183 <p> 184 <span class="blue">Max = 0ε (Mean = 0ε)</span> 185 </p> 186 </td> 187<td> 188 <p> 189 <span class="blue">Max = 1.97ε (Mean = 0.757ε)</span> 190 </p> 191 </td> 192<td> 193 <p> 194 <span class="blue">Max = 1.97ε (Mean = 0.757ε)</span> 195 </p> 196 </td> 197<td> 198 <p> 199 <span class="blue">Max = 1.36ε (Mean = 0.782ε)</span> 200 </p> 201 </td> 202</tr> 203<tr> 204<td> 205 <p> 206 Bessel I'n: Mathworld Data (Integer Version) 207 </p> 208 </td> 209<td> 210 <p> 211 <span class="blue">Max = 0ε (Mean = 0ε)</span> 212 </p> 213 </td> 214<td> 215 <p> 216 <span class="blue">Max = 2.31ε (Mean = 1.41ε)</span> 217 </p> 218 </td> 219<td> 220 <p> 221 <span class="blue">Max = 701ε (Mean = 212ε)</span> 222 </p> 223 </td> 224<td> 225 <p> 226 <span class="blue">Max = 3.61ε (Mean = 1.22ε)</span> 227 </p> 228 </td> 229</tr> 230</tbody> 231</table></div> 232</div> 233<br class="table-break"><div class="table"> 234<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime"></a><p class="title"><b>Table 8.51. Error rates for cyl_bessel_i_prime</b></p> 235<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i_prime"> 236<colgroup> 237<col> 238<col> 239<col> 240<col> 241<col> 242</colgroup> 243<thead><tr> 244<th> 245 </th> 246<th> 247 <p> 248 GNU C++ version 7.1.0<br> linux<br> double 249 </p> 250 </th> 251<th> 252 <p> 253 GNU C++ version 7.1.0<br> linux<br> long double 254 </p> 255 </th> 256<th> 257 <p> 258 Sun compiler version 0x5150<br> Sun Solaris<br> long double 259 </p> 260 </th> 261<th> 262 <p> 263 Microsoft Visual C++ version 14.1<br> Win32<br> double 264 </p> 265 </th> 266</tr></thead> 267<tbody> 268<tr> 269<td> 270 <p> 271 Bessel I'0: Mathworld Data 272 </p> 273 </td> 274<td> 275 <p> 276 <span class="blue">Max = 0ε (Mean = 0ε)</span> 277 </p> 278 </td> 279<td> 280 <p> 281 <span class="blue">Max = 0.82ε (Mean = 0.259ε)</span> 282 </p> 283 </td> 284<td> 285 <p> 286 <span class="blue">Max = 0.82ε (Mean = 0.259ε)</span> 287 </p> 288 </td> 289<td> 290 <p> 291 <span class="blue">Max = 0.82ε (Mean = 0.354ε)</span> 292 </p> 293 </td> 294</tr> 295<tr> 296<td> 297 <p> 298 Bessel I'1: Mathworld Data 299 </p> 300 </td> 301<td> 302 <p> 303 <span class="blue">Max = 0ε (Mean = 0ε)</span> 304 </p> 305 </td> 306<td> 307 <p> 308 <span class="blue">Max = 1.97ε (Mean = 0.757ε)</span> 309 </p> 310 </td> 311<td> 312 <p> 313 <span class="blue">Max = 1.97ε (Mean = 0.757ε)</span> 314 </p> 315 </td> 316<td> 317 <p> 318 <span class="blue">Max = 1.36ε (Mean = 0.782ε)</span> 319 </p> 320 </td> 321</tr> 322<tr> 323<td> 324 <p> 325 Bessel I'n: Mathworld Data 326 </p> 327 </td> 328<td> 329 <p> 330 <span class="blue">Max = 0ε (Mean = 0ε)</span> 331 </p> 332 </td> 333<td> 334 <p> 335 <span class="blue">Max = 2.31ε (Mean = 1.41ε)</span> 336 </p> 337 </td> 338<td> 339 <p> 340 <span class="blue">Max = 701ε (Mean = 212ε)</span> 341 </p> 342 </td> 343<td> 344 <p> 345 <span class="blue">Max = 3.61ε (Mean = 1.22ε)</span> 346 </p> 347 </td> 348</tr> 349<tr> 350<td> 351 <p> 352 Bessel I'v: Mathworld Data 353 </p> 354 </td> 355<td> 356 <p> 357 <span class="blue">Max = 1.62ε (Mean = 0.512ε)</span> 358 </p> 359 </td> 360<td> 361 <p> 362 <span class="blue">Max = 2.89e+03ε (Mean = 914ε)</span> 363 </p> 364 </td> 365<td> 366 <p> 367 <span class="blue">Max = 2.89e+03ε (Mean = 914ε)</span> 368 </p> 369 </td> 370<td> 371 <p> 372 <span class="blue">Max = 3.76e+03ε (Mean = 1.19e+03ε)</span> 373 </p> 374 </td> 375</tr> 376<tr> 377<td> 378 <p> 379 Bessel I'n: Random Data 380 </p> 381 </td> 382<td> 383 <p> 384 <span class="blue">Max = 0ε (Mean = 0ε)</span> 385 </p> 386 </td> 387<td> 388 <p> 389 <span class="blue">Max = 3.95ε (Mean = 1.06ε)</span> 390 </p> 391 </td> 392<td> 393 <p> 394 <span class="blue">Max = 195ε (Mean = 37.1ε)</span> 395 </p> 396 </td> 397<td> 398 <p> 399 <span class="blue">Max = 9.85ε (Mean = 1.82ε)</span> 400 </p> 401 </td> 402</tr> 403<tr> 404<td> 405 <p> 406 Bessel I'v: Random Data 407 </p> 408 </td> 409<td> 410 <p> 411 <span class="blue">Max = 0ε (Mean = 0ε)</span> 412 </p> 413 </td> 414<td> 415 <p> 416 <span class="blue">Max = 14.1ε (Mean = 2.93ε)</span> 417 </p> 418 </td> 419<td> 420 <p> 421 <span class="blue">Max = 336ε (Mean = 68.5ε)</span> 422 </p> 423 </td> 424<td> 425 <p> 426 <span class="blue">Max = 14ε (Mean = 2.5ε)</span> 427 </p> 428 </td> 429</tr> 430<tr> 431<td> 432 <p> 433 Bessel I'v: Mathworld Data (large values) 434 </p> 435 </td> 436<td> 437 <p> 438 <span class="blue">Max = 0ε (Mean = 0ε)</span> 439 </p> 440 </td> 441<td> 442 <p> 443 <span class="blue">Max = 42.6ε (Mean = 20.2ε)</span> 444 </p> 445 </td> 446<td> 447 <p> 448 <span class="blue">Max = 42.6ε (Mean = 20.2ε)</span> 449 </p> 450 </td> 451<td> 452 <p> 453 <span class="blue">Max = 59.5ε (Mean = 26.6ε)</span> 454 </p> 455 </td> 456</tr> 457</tbody> 458</table></div> 459</div> 460<br class="table-break"><div class="table"> 461<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime_integer_orders_"></a><p class="title"><b>Table 8.52. Error rates for cyl_bessel_j_prime (integer orders)</b></p> 462<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j_prime (integer orders)"> 463<colgroup> 464<col> 465<col> 466<col> 467<col> 468<col> 469</colgroup> 470<thead><tr> 471<th> 472 </th> 473<th> 474 <p> 475 GNU C++ version 7.1.0<br> linux<br> double 476 </p> 477 </th> 478<th> 479 <p> 480 GNU C++ version 7.1.0<br> linux<br> long double 481 </p> 482 </th> 483<th> 484 <p> 485 Sun compiler version 0x5150<br> Sun Solaris<br> long double 486 </p> 487 </th> 488<th> 489 <p> 490 Microsoft Visual C++ version 14.1<br> Win32<br> double 491 </p> 492 </th> 493</tr></thead> 494<tbody> 495<tr> 496<td> 497 <p> 498 Bessel J0': Mathworld Data (Integer Version) 499 </p> 500 </td> 501<td> 502 <p> 503 <span class="blue">Max = 0ε (Mean = 0ε)</span> 504 </p> 505 </td> 506<td> 507 <p> 508 <span class="blue">Max = 18.9ε (Mean = 6.82ε)</span> 509 </p> 510 </td> 511<td> 512 <p> 513 <span class="blue">Max = 18.9ε (Mean = 6.72ε)</span> 514 </p> 515 </td> 516<td> 517 <p> 518 <span class="blue">Max = 6.62ε (Mean = 2.55ε)</span> 519 </p> 520 </td> 521</tr> 522<tr> 523<td> 524 <p> 525 Bessel J0': Mathworld Data (Tricky cases) (Integer Version) 526 </p> 527 </td> 528<td> 529 <p> 530 <span class="blue">Max = 0ε (Mean = 0ε)</span> 531 </p> 532 </td> 533<td> 534 <p> 535 <span class="blue">Max = 7.44ε (Mean = 3.34ε)</span> 536 </p> 537 </td> 538<td> 539 <p> 540 <span class="blue">Max = 7.44ε (Mean = 3.31ε)</span> 541 </p> 542 </td> 543<td> 544 <p> 545 <span class="blue">Max = 3.67ε (Mean = 1.74ε)</span> 546 </p> 547 </td> 548</tr> 549<tr> 550<td> 551 <p> 552 Bessel J1': Mathworld Data (Integer Version) 553 </p> 554 </td> 555<td> 556 <p> 557 <span class="blue">Max = 0ε (Mean = 0ε)</span> 558 </p> 559 </td> 560<td> 561 <p> 562 <span class="blue">Max = 7.9ε (Mean = 3.37ε)</span> 563 </p> 564 </td> 565<td> 566 <p> 567 <span class="blue">Max = 7.9ε (Mean = 3.37ε)</span> 568 </p> 569 </td> 570<td> 571 <p> 572 <span class="blue">Max = 0.999ε (Mean = 0.627ε)</span> 573 </p> 574 </td> 575</tr> 576<tr> 577<td> 578 <p> 579 Bessel J1': Mathworld Data (tricky cases) (Integer Version) 580 </p> 581 </td> 582<td> 583 <p> 584 <span class="blue">Max = 287ε (Mean = 129ε)</span> 585 </p> 586 </td> 587<td> 588 <p> 589 <span class="blue">Max = 5.88e+05ε (Mean = 2.63e+05ε)</span> 590 </p> 591 </td> 592<td> 593 <p> 594 <span class="blue">Max = 5.88e+05ε (Mean = 2.63e+05ε)</span> 595 </p> 596 </td> 597<td> 598 <p> 599 <span class="blue">Max = 288ε (Mean = 129ε)</span> 600 </p> 601 </td> 602</tr> 603<tr> 604<td> 605 <p> 606 Bessel JN': Mathworld Data (Integer Version) 607 </p> 608 </td> 609<td> 610 <p> 611 <span class="blue">Max = 0.527ε (Mean = 0.128ε)</span> 612 </p> 613 </td> 614<td> 615 <p> 616 <span class="blue">Max = 1.29e+03ε (Mean = 312ε)</span> 617 </p> 618 </td> 619<td> 620 <p> 621 <span class="blue">Max = 1.29e+03ε (Mean = 355ε)</span> 622 </p> 623 </td> 624<td> 625 <p> 626 <span class="blue">Max = 14ε (Mean = 6.13ε)</span> 627 </p> 628 </td> 629</tr> 630</tbody> 631</table></div> 632</div> 633<br class="table-break"><div class="table"> 634<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime"></a><p class="title"><b>Table 8.53. Error rates for cyl_bessel_j_prime</b></p> 635<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j_prime"> 636<colgroup> 637<col> 638<col> 639<col> 640<col> 641<col> 642</colgroup> 643<thead><tr> 644<th> 645 </th> 646<th> 647 <p> 648 GNU C++ version 7.1.0<br> linux<br> double 649 </p> 650 </th> 651<th> 652 <p> 653 GNU C++ version 7.1.0<br> linux<br> long double 654 </p> 655 </th> 656<th> 657 <p> 658 Sun compiler version 0x5150<br> Sun Solaris<br> long double 659 </p> 660 </th> 661<th> 662 <p> 663 Microsoft Visual C++ version 14.1<br> Win32<br> double 664 </p> 665 </th> 666</tr></thead> 667<tbody> 668<tr> 669<td> 670 <p> 671 Bessel J0': Mathworld Data 672 </p> 673 </td> 674<td> 675 <p> 676 <span class="blue">Max = 0ε (Mean = 0ε)</span> 677 </p> 678 </td> 679<td> 680 <p> 681 <span class="blue">Max = 18.9ε (Mean = 6.82ε)</span> 682 </p> 683 </td> 684<td> 685 <p> 686 <span class="blue">Max = 18.9ε (Mean = 6.72ε)</span> 687 </p> 688 </td> 689<td> 690 <p> 691 <span class="blue">Max = 6.62ε (Mean = 2.55ε)</span> 692 </p> 693 </td> 694</tr> 695<tr> 696<td> 697 <p> 698 Bessel J0': Mathworld Data (Tricky cases) 699 </p> 700 </td> 701<td> 702 <p> 703 <span class="blue">Max = 0ε (Mean = 0ε)</span> 704 </p> 705 </td> 706<td> 707 <p> 708 <span class="blue">Max = 7.44ε (Mean = 3.34ε)</span> 709 </p> 710 </td> 711<td> 712 <p> 713 <span class="blue">Max = 7.44ε (Mean = 3.31ε)</span> 714 </p> 715 </td> 716<td> 717 <p> 718 <span class="blue">Max = 3.67ε (Mean = 1.74ε)</span> 719 </p> 720 </td> 721</tr> 722<tr> 723<td> 724 <p> 725 Bessel J1': Mathworld Data 726 </p> 727 </td> 728<td> 729 <p> 730 <span class="blue">Max = 0ε (Mean = 0ε)</span> 731 </p> 732 </td> 733<td> 734 <p> 735 <span class="blue">Max = 7.9ε (Mean = 3.37ε)</span> 736 </p> 737 </td> 738<td> 739 <p> 740 <span class="blue">Max = 7.9ε (Mean = 3.37ε)</span> 741 </p> 742 </td> 743<td> 744 <p> 745 <span class="blue">Max = 0.999ε (Mean = 0.627ε)</span> 746 </p> 747 </td> 748</tr> 749<tr> 750<td> 751 <p> 752 Bessel J1': Mathworld Data (tricky cases) 753 </p> 754 </td> 755<td> 756 <p> 757 <span class="blue">Max = 287ε (Mean = 129ε)</span> 758 </p> 759 </td> 760<td> 761 <p> 762 <span class="blue">Max = 5.88e+05ε (Mean = 2.63e+05ε)</span> 763 </p> 764 </td> 765<td> 766 <p> 767 <span class="blue">Max = 5.88e+05ε (Mean = 2.63e+05ε)</span> 768 </p> 769 </td> 770<td> 771 <p> 772 <span class="blue">Max = 288ε (Mean = 129ε)</span> 773 </p> 774 </td> 775</tr> 776<tr> 777<td> 778 <p> 779 Bessel JN': Mathworld Data 780 </p> 781 </td> 782<td> 783 <p> 784 <span class="blue">Max = 0.527ε (Mean = 0.128ε)</span> 785 </p> 786 </td> 787<td> 788 <p> 789 <span class="blue">Max = 1.29e+03ε (Mean = 312ε)</span> 790 </p> 791 </td> 792<td> 793 <p> 794 <span class="blue">Max = 1.29e+03ε (Mean = 355ε)</span> 795 </p> 796 </td> 797<td> 798 <p> 799 <span class="blue">Max = 14ε (Mean = 6.13ε)</span> 800 </p> 801 </td> 802</tr> 803<tr> 804<td> 805 <p> 806 Bessel J': Mathworld Data 807 </p> 808 </td> 809<td> 810 <p> 811 <span class="blue">Max = 21.5ε (Mean = 4.7ε)</span> 812 </p> 813 </td> 814<td> 815 <p> 816 <span class="blue">Max = 42.5ε (Mean = 9.31ε)</span> 817 </p> 818 </td> 819<td> 820 <p> 821 <span class="blue">Max = 42.5ε (Mean = 9.32ε)</span> 822 </p> 823 </td> 824<td> 825 <p> 826 <span class="blue">Max = 23.7ε (Mean = 8ε)</span> 827 </p> 828 </td> 829</tr> 830<tr> 831<td> 832 <p> 833 Bessel J': Mathworld Data (large values) 834 </p> 835 </td> 836<td> 837 <p> 838 <span class="blue">Max = 0ε (Mean = 0ε)</span> 839 </p> 840 </td> 841<td> 842 <p> 843 <span class="blue">Max = 989ε (Mean = 495ε)</span> 844 </p> 845 </td> 846<td> 847 <p> 848 <span class="blue">Max = 989ε (Mean = 495ε)</span> 849 </p> 850 </td> 851<td> 852 <p> 853 <span class="blue">Max = 2.9ε (Mean = 1.61ε)</span> 854 </p> 855 </td> 856</tr> 857<tr> 858<td> 859 <p> 860 Bessel JN': Random Data 861 </p> 862 </td> 863<td> 864 <p> 865 <span class="blue">Max = 0.593ε (Mean = 0.0396ε)</span> 866 </p> 867 </td> 868<td> 869 <p> 870 <span class="blue">Max = 11.3ε (Mean = 1.85ε)</span> 871 </p> 872 </td> 873<td> 874 <p> 875 <span class="blue">Max = 79.4ε (Mean = 16.2ε)</span> 876 </p> 877 </td> 878<td> 879 <p> 880 <span class="blue">Max = 6.34ε (Mean = 0.999ε)</span> 881 </p> 882 </td> 883</tr> 884<tr> 885<td> 886 <p> 887 Bessel J': Random Data 888 </p> 889 </td> 890<td> 891 <p> 892 <span class="blue">Max = 0.885ε (Mean = 0.033ε)</span> 893 </p> 894 </td> 895<td> 896 <p> 897 <span class="blue">Max = 139ε (Mean = 6.47ε)</span> 898 </p> 899 </td> 900<td> 901 <p> 902 <span class="blue">Max = 279ε (Mean = 27.2ε)</span> 903 </p> 904 </td> 905<td> 906 <p> 907 <span class="blue">Max = 176ε (Mean = 9.75ε)</span> 908 </p> 909 </td> 910</tr> 911<tr> 912<td> 913 <p> 914 Bessel J': Random Data (Tricky large values) 915 </p> 916 </td> 917<td> 918 <p> 919 <span class="blue">Max = 0ε (Mean = 0ε)</span> 920 </p> 921 </td> 922<td> 923 <p> 924 <span class="blue">Max = 474ε (Mean = 62.2ε)</span> 925 </p> 926 </td> 927<td> 928 <p> 929 <span class="blue">Max = 474ε (Mean = 64.5ε)</span> 930 </p> 931 </td> 932<td> 933 <p> 934 <span class="blue">Max = 379ε (Mean = 45.4ε)</span> 935 </p> 936 </td> 937</tr> 938</tbody> 939</table></div> 940</div> 941<br class="table-break"><div class="table"> 942<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime_integer_orders_"></a><p class="title"><b>Table 8.54. Error rates for cyl_bessel_k_prime (integer orders)</b></p> 943<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k_prime (integer orders)"> 944<colgroup> 945<col> 946<col> 947<col> 948<col> 949<col> 950</colgroup> 951<thead><tr> 952<th> 953 </th> 954<th> 955 <p> 956 GNU C++ version 7.1.0<br> linux<br> double 957 </p> 958 </th> 959<th> 960 <p> 961 GNU C++ version 7.1.0<br> linux<br> long double 962 </p> 963 </th> 964<th> 965 <p> 966 Sun compiler version 0x5150<br> Sun Solaris<br> long double 967 </p> 968 </th> 969<th> 970 <p> 971 Microsoft Visual C++ version 14.1<br> Win32<br> double 972 </p> 973 </th> 974</tr></thead> 975<tbody> 976<tr> 977<td> 978 <p> 979 Bessel K'0: Mathworld Data (Integer Version) 980 </p> 981 </td> 982<td> 983 <p> 984 <span class="blue">Max = 0ε (Mean = 0ε)</span> 985 </p> 986 </td> 987<td> 988 <p> 989 <span class="blue">Max = 0.786ε (Mean = 0.329ε)</span> 990 </p> 991 </td> 992<td> 993 <p> 994 <span class="blue">Max = 0.786ε (Mean = 0.329ε)</span> 995 </p> 996 </td> 997<td> 998 <p> 999 <span class="blue">Max = 0.786ε (Mean = 0.39ε)</span> 1000 </p> 1001 </td> 1002</tr> 1003<tr> 1004<td> 1005 <p> 1006 Bessel K'1: Mathworld Data (Integer Version) 1007 </p> 1008 </td> 1009<td> 1010 <p> 1011 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1012 </p> 1013 </td> 1014<td> 1015 <p> 1016 <span class="blue">Max = 0.736ε (Mean = 0.389ε)</span> 1017 </p> 1018 </td> 1019<td> 1020 <p> 1021 <span class="blue">Max = 0.736ε (Mean = 0.389ε)</span> 1022 </p> 1023 </td> 1024<td> 1025 <p> 1026 <span class="blue">Max = 0.761ε (Mean = 0.444ε)</span> 1027 </p> 1028 </td> 1029</tr> 1030<tr> 1031<td> 1032 <p> 1033 Bessel K'n: Mathworld Data (Integer Version) 1034 </p> 1035 </td> 1036<td> 1037 <p> 1038 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1039 </p> 1040 </td> 1041<td> 1042 <p> 1043 <span class="blue">Max = 2.16ε (Mean = 1.08ε)</span> 1044 </p> 1045 </td> 1046<td> 1047 <p> 1048 <span class="blue">Max = 2.16ε (Mean = 1.08ε)</span> 1049 </p> 1050 </td> 1051<td> 1052 <p> 1053 <span class="blue">Max = 4.17ε (Mean = 1.75ε)</span> 1054 </p> 1055 </td> 1056</tr> 1057</tbody> 1058</table></div> 1059</div> 1060<br class="table-break"><div class="table"> 1061<a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime"></a><p class="title"><b>Table 8.55. Error rates for cyl_bessel_k_prime</b></p> 1062<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k_prime"> 1063<colgroup> 1064<col> 1065<col> 1066<col> 1067<col> 1068<col> 1069</colgroup> 1070<thead><tr> 1071<th> 1072 </th> 1073<th> 1074 <p> 1075 GNU C++ version 7.1.0<br> linux<br> double 1076 </p> 1077 </th> 1078<th> 1079 <p> 1080 GNU C++ version 7.1.0<br> linux<br> long double 1081 </p> 1082 </th> 1083<th> 1084 <p> 1085 Sun compiler version 0x5150<br> Sun Solaris<br> long double 1086 </p> 1087 </th> 1088<th> 1089 <p> 1090 Microsoft Visual C++ version 14.1<br> Win32<br> double 1091 </p> 1092 </th> 1093</tr></thead> 1094<tbody> 1095<tr> 1096<td> 1097 <p> 1098 Bessel K'0: Mathworld Data 1099 </p> 1100 </td> 1101<td> 1102 <p> 1103 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1104 </p> 1105 </td> 1106<td> 1107 <p> 1108 <span class="blue">Max = 0.786ε (Mean = 0.329ε)</span> 1109 </p> 1110 </td> 1111<td> 1112 <p> 1113 <span class="blue">Max = 0.786ε (Mean = 0.329ε)</span> 1114 </p> 1115 </td> 1116<td> 1117 <p> 1118 <span class="blue">Max = 0.786ε (Mean = 0.39ε)</span> 1119 </p> 1120 </td> 1121</tr> 1122<tr> 1123<td> 1124 <p> 1125 Bessel K'1: Mathworld Data 1126 </p> 1127 </td> 1128<td> 1129 <p> 1130 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1131 </p> 1132 </td> 1133<td> 1134 <p> 1135 <span class="blue">Max = 0.736ε (Mean = 0.389ε)</span> 1136 </p> 1137 </td> 1138<td> 1139 <p> 1140 <span class="blue">Max = 0.736ε (Mean = 0.389ε)</span> 1141 </p> 1142 </td> 1143<td> 1144 <p> 1145 <span class="blue">Max = 0.761ε (Mean = 0.444ε)</span> 1146 </p> 1147 </td> 1148</tr> 1149<tr> 1150<td> 1151 <p> 1152 Bessel K'n: Mathworld Data 1153 </p> 1154 </td> 1155<td> 1156 <p> 1157 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1158 </p> 1159 </td> 1160<td> 1161 <p> 1162 <span class="blue">Max = 2.16ε (Mean = 1.08ε)</span> 1163 </p> 1164 </td> 1165<td> 1166 <p> 1167 <span class="blue">Max = 2.16ε (Mean = 1.08ε)</span> 1168 </p> 1169 </td> 1170<td> 1171 <p> 1172 <span class="blue">Max = 4.17ε (Mean = 1.75ε)</span> 1173 </p> 1174 </td> 1175</tr> 1176<tr> 1177<td> 1178 <p> 1179 Bessel K'v: Mathworld Data 1180 </p> 1181 </td> 1182<td> 1183 <p> 1184 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1185 </p> 1186 </td> 1187<td> 1188 <p> 1189 <span class="blue">Max = 3.94ε (Mean = 2.44ε)</span> 1190 </p> 1191 </td> 1192<td> 1193 <p> 1194 <span class="blue">Max = 3.94ε (Mean = 2.34ε)</span> 1195 </p> 1196 </td> 1197<td> 1198 <p> 1199 <span class="blue">Max = 3.94ε (Mean = 1.47ε)</span> 1200 </p> 1201 </td> 1202</tr> 1203<tr> 1204<td> 1205 <p> 1206 Bessel K'v: Mathworld Data (large values) 1207 </p> 1208 </td> 1209<td> 1210 <p> 1211 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1212 </p> 1213 </td> 1214<td> 1215 <p> 1216 <span class="blue">Max = 59.2ε (Mean = 42.9ε)</span> 1217 </p> 1218 </td> 1219<td> 1220 <p> 1221 <span class="blue">Max = 58.7ε (Mean = 42.6ε)</span> 1222 </p> 1223 </td> 1224<td> 1225 <p> 1226 <span class="blue">Max = 18.6ε (Mean = 11.8ε)</span> 1227 </p> 1228 </td> 1229</tr> 1230<tr> 1231<td> 1232 <p> 1233 Bessel K'n: Random Data 1234 </p> 1235 </td> 1236<td> 1237 <p> 1238 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1239 </p> 1240 </td> 1241<td> 1242 <p> 1243 <span class="blue">Max = 4.45ε (Mean = 1.19ε)</span> 1244 </p> 1245 </td> 1246<td> 1247 <p> 1248 <span class="blue">Max = 4.45ε (Mean = 1.19ε)</span> 1249 </p> 1250 </td> 1251<td> 1252 <p> 1253 <span class="blue">Max = 9.67ε (Mean = 1.73ε)</span> 1254 </p> 1255 </td> 1256</tr> 1257<tr> 1258<td> 1259 <p> 1260 Bessel K'v: Random Data 1261 </p> 1262 </td> 1263<td> 1264 <p> 1265 <span class="blue">Max = 0ε (Mean = 0ε)</span> 1266 </p> 1267 </td> 1268<td> 1269 <p> 1270 <span class="blue">Max = 7.95ε (Mean = 1.53ε)</span> 1271 </p> 1272 </td> 1273<td> 1274 <p> 1275 <span class="blue">Max = 7.95ε (Mean = 1.52ε)</span> 1276 </p> 1277 </td> 1278<td> 1279 <p> 1280 <span class="blue">Max = 8.32ε (Mean = 1.65ε)</span> 1281 </p> 1282 </td> 1283</tr> 1284</tbody> 1285</table></div> 1286</div> 1287<br class="table-break"><div class="table"> 1288<a name="math_toolkit.bessel.bessel_derivatives.table_sph_bessel_prime"></a><p class="title"><b>Table 8.56. Error rates for sph_bessel_prime</b></p> 1289<div class="table-contents"><table class="table" summary="Error rates for sph_bessel_prime"> 1290<colgroup> 1291<col> 1292<col> 1293<col> 1294<col> 1295<col> 1296</colgroup> 1297<thead><tr> 1298<th> 1299 </th> 1300<th> 1301 <p> 1302 GNU C++ version 7.1.0<br> linux<br> double 1303 </p> 1304 </th> 1305<th> 1306 <p> 1307 GNU C++ version 7.1.0<br> linux<br> long double 1308 </p> 1309 </th> 1310<th> 1311 <p> 1312 Sun compiler version 0x5150<br> Sun Solaris<br> long double 1313 </p> 1314 </th> 1315<th> 1316 <p> 1317 Microsoft Visual C++ version 14.1<br> Win32<br> double 1318 </p> 1319 </th> 1320</tr></thead> 1321<tbody><tr> 1322<td> 1323 <p> 1324 Bessel j': Random Data 1325 </p> 1326 </td> 1327<td> 1328 <p> 1329 <span class="blue">Max = 0.753ε (Mean = 0.0343ε)</span> 1330 </p> 1331 </td> 1332<td> 1333 <p> 1334 <span class="blue">Max = 167ε (Mean = 12ε)</span> 1335 </p> 1336 </td> 1337<td> 1338 <p> 1339 <span class="blue">Max = 167ε (Mean = 33.2ε)</span> 1340 </p> 1341 </td> 1342<td> 1343 <p> 1344 <span class="blue">Max = 307ε (Mean = 25.2ε)</span> 1345 </p> 1346 </td> 1347</tr></tbody> 1348</table></div> 1349</div> 1350<br class="table-break"><div class="table"> 1351<a name="math_toolkit.bessel.bessel_derivatives.table_sph_neumann_prime"></a><p class="title"><b>Table 8.57. Error rates for sph_neumann_prime</b></p> 1352<div class="table-contents"><table class="table" summary="Error rates for sph_neumann_prime"> 1353<colgroup> 1354<col> 1355<col> 1356<col> 1357<col> 1358<col> 1359</colgroup> 1360<thead><tr> 1361<th> 1362 </th> 1363<th> 1364 <p> 1365 GNU C++ version 7.1.0<br> linux<br> double 1366 </p> 1367 </th> 1368<th> 1369 <p> 1370 GNU C++ version 7.1.0<br> linux<br> long double 1371 </p> 1372 </th> 1373<th> 1374 <p> 1375 Sun compiler version 0x5150<br> Sun Solaris<br> long double 1376 </p> 1377 </th> 1378<th> 1379 <p> 1380 Microsoft Visual C++ version 14.1<br> Win32<br> double 1381 </p> 1382 </th> 1383</tr></thead> 1384<tbody><tr> 1385<td> 1386 <p> 1387 y': Random Data 1388 </p> 1389 </td> 1390<td> 1391 <p> 1392 <span class="blue">Max = 0.988ε (Mean = 0.0869ε)</span> 1393 </p> 1394 </td> 1395<td> 1396 <p> 1397 <span class="blue">Max = 158ε (Mean = 18.8ε)</span> 1398 </p> 1399 </td> 1400<td> 1401 <p> 1402 <span class="blue">Max = 158ε (Mean = 20.2ε)</span> 1403 </p> 1404 </td> 1405<td> 1406 <p> 1407 <span class="blue">Max = 296ε (Mean = 25.6ε)</span> 1408 </p> 1409 </td> 1410</tr></tbody> 1411</table></div> 1412</div> 1413<br class="table-break"><h5> 1414<a name="math_toolkit.bessel.bessel_derivatives.h4"></a> 1415 <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.implementation"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.implementation">Implementation</a> 1416 </h5> 1417<p> 1418 In the general case, the derivatives are calculated using the relations: 1419 </p> 1420<div class="blockquote"><blockquote class="blockquote"><p> 1421 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives1.svg"></span> 1422 1423 </p></blockquote></div> 1424<p> 1425 There are also a number of special cases, for large x we have: 1426 </p> 1427<div class="blockquote"><blockquote class="blockquote"><p> 1428 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives4.svg"></span> 1429 1430 </p></blockquote></div> 1431<p> 1432 And for small x: 1433 </p> 1434<div class="blockquote"><blockquote class="blockquote"><p> 1435 <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives5.svg"></span> 1436 1437 </p></blockquote></div> 1438</div> 1439<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 1440<td align="left"></td> 1441<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 1442 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 1443 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 1444 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 1445 Daryle Walker and Xiaogang Zhang<p> 1446 Distributed under the Boost Software License, Version 1.0. (See accompanying 1447 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>) 1448 </p> 1449</div></td> 1450</tr></table> 1451<hr> 1452<div class="spirit-nav"> 1453<a accesskey="p" href="sph_bessel.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="../hankel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 1454</div> 1455</body> 1456</html> 1457
