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61<h1>Bessel Functions<br>
62<small>
63[<a class="el" href="group__base.html">Base Module</a>]</small>
64</h1><table border="0" cellpadding="0" cellspacing="0">
65<tr><td></td></tr>
66<tr><td colspan="2"><br><h2>Functions</h2></td></tr>
67<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#ga25e070e28f322ad205c8251f0dd6e1d">itpp::besselj</a> (int nu, double x)</td></tr>
68
69<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of first kind of order <em>nu</em> for <em>nu</em> integer.  <a href="#ga25e070e28f322ad205c8251f0dd6e1d"></a><br></td></tr>
70<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g7d19aee09bfd9c7a64fd0a076a2d22b7"></a><!-- doxytag: member="besselfunctions::besselj" ref="g7d19aee09bfd9c7a64fd0a076a2d22b7" args="(int nu, const vec &amp;x)" -->
71<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g7d19aee09bfd9c7a64fd0a076a2d22b7">itpp::besselj</a> (int nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
72
73<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of first kind of order <em>nu</em> for <em>nu</em> integer. <br></td></tr>
74<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="ga66dd30e4d556696cddb9236459f43c1"></a><!-- doxytag: member="besselfunctions::besselj" ref="ga66dd30e4d556696cddb9236459f43c1" args="(double nu, double x)" -->
75double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#ga66dd30e4d556696cddb9236459f43c1">itpp::besselj</a> (double nu, double x)</td></tr>
76
77<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of first kind of order <em>nu</em>. <em>nu</em> is real. <br></td></tr>
78<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g595f0bad25b9175452ade5eff56d6a68"></a><!-- doxytag: member="besselfunctions::besselj" ref="g595f0bad25b9175452ade5eff56d6a68" args="(double nu, const vec &amp;x)" -->
79<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g595f0bad25b9175452ade5eff56d6a68">itpp::besselj</a> (double nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
80
81<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of first kind of order <em>nu</em>. <em>nu</em> is real. <br></td></tr>
82<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g15832f12203d675afad32e69670130dd">itpp::bessely</a> (int nu, double x)</td></tr>
83
84<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of second kind of order <em>nu</em>. <em>nu</em> is integer.  <a href="#g15832f12203d675afad32e69670130dd"></a><br></td></tr>
85<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g3ba7923e64d809067429ded731b1fb73"></a><!-- doxytag: member="besselfunctions::bessely" ref="g3ba7923e64d809067429ded731b1fb73" args="(int nu, const vec &amp;x)" -->
86<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g3ba7923e64d809067429ded731b1fb73">itpp::bessely</a> (int nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
87
88<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of second kind of order <em>nu</em>. <em>nu</em> is integer. <br></td></tr>
89<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g7c9b620dab32b456a92dd8e69b9af92e"></a><!-- doxytag: member="besselfunctions::bessely" ref="g7c9b620dab32b456a92dd8e69b9af92e" args="(double nu, double x)" -->
90double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g7c9b620dab32b456a92dd8e69b9af92e">itpp::bessely</a> (double nu, double x)</td></tr>
91
92<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of second kind of order <em>nu</em>. <em>nu</em> is real. <br></td></tr>
93<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g8a42390a681660010bd21eda1c3a497e"></a><!-- doxytag: member="besselfunctions::bessely" ref="g8a42390a681660010bd21eda1c3a497e" args="(double nu, const vec &amp;x)" -->
94<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g8a42390a681660010bd21eda1c3a497e">itpp::bessely</a> (double nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
95
96<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Bessel function of second kind of order <em>nu</em>. <em>nu</em> is real. <br></td></tr>
97<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g44e844761515d3dccd07ef7be23423cf">itpp::besseli</a> (double nu, double x)</td></tr>
98
99<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Modified Bessel function of first kind of order <em>nu</em>. <em>nu</em> is <em>double</em>. <em>x</em> is <em>double</em>.  <a href="#g44e844761515d3dccd07ef7be23423cf"></a><br></td></tr>
100<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g106e55fe4ffff36a29f62991dc18b07d"></a><!-- doxytag: member="besselfunctions::besseli" ref="g106e55fe4ffff36a29f62991dc18b07d" args="(double nu, const vec &amp;x)" -->
101<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g106e55fe4ffff36a29f62991dc18b07d">itpp::besseli</a> (double nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
102
103<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Modified Bessel function of first kind of order <em>nu</em>. <em>nu</em> is <em>double</em>. <em>x</em> is <em>double</em>. <br></td></tr>
104<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#gff9230fd225656cdf47c384cb7dc37de">itpp::besselk</a> (int nu, double x)</td></tr>
105
106<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Modified Bessel function of second kind of order <em>nu</em>. <em>nu</em> is double. <em>x</em> is double.  <a href="#gff9230fd225656cdf47c384cb7dc37de"></a><br></td></tr>
107<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g33c2f08f8475febf53e26598f4bc3efc"></a><!-- doxytag: member="besselfunctions::besselk" ref="g33c2f08f8475febf53e26598f4bc3efc" args="(int nu, const vec &amp;x)" -->
108<a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a>&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__besselfunctions.html#g33c2f08f8475febf53e26598f4bc3efc">itpp::besselk</a> (int nu, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
109
110<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Modified Bessel function of second kind of order <em>nu</em>. <em>nu</em> is double. <em>x</em> is double. <br></td></tr>
111</table>
112<hr><a name="_details"></a><h2>Detailed Description</h2>
113end of math group <hr><h2>Function Documentation</h2>
114<a class="anchor" name="g44e844761515d3dccd07ef7be23423cf"></a><!-- doxytag: member="itpp::besseli" ref="g44e844761515d3dccd07ef7be23423cf" args="(double nu, double x)" -->
115<div class="memitem">
116<div class="memproto">
117      <table class="memname">
118        <tr>
119          <td class="memname">double itpp::besseli           </td>
120          <td>(</td>
121          <td class="paramtype">double&nbsp;</td>
122          <td class="paramname"> <em>nu</em>, </td>
123        </tr>
124        <tr>
125          <td class="paramkey"></td>
126          <td></td>
127          <td class="paramtype">double&nbsp;</td>
128          <td class="paramname"> <em>x</em></td><td>&nbsp;</td>
129        </tr>
130        <tr>
131          <td></td>
132          <td>)</td>
133          <td></td><td></td><td></td>
134        </tr>
135      </table>
136</div>
137<div class="memdoc">
138
139<p>
140Modified Bessel function of first kind of order <em>nu</em>. <em>nu</em> is <em>double</em>. <em>x</em> is <em>double</em>.
141<p>
142The Modified Bessel function of first kind is defined as: <p class="formulaDsp">
143<img class="formulaDsp" alt="\[ I_{\nu}(x) = i^{-\nu} J_{\nu}(ix) \]" src="form_175.png">
144<p>
145 where <img class="formulaInl" alt="$\nu$" src="form_172.png"> is the order and <img class="formulaInl" alt="$ 0 < x < \infty $" src="form_173.png">.
146</div>
147</div><p>
148<a class="anchor" name="ga25e070e28f322ad205c8251f0dd6e1d"></a><!-- doxytag: member="itpp::besselj" ref="ga25e070e28f322ad205c8251f0dd6e1d" args="(int nu, double x)" -->
149<div class="memitem">
150<div class="memproto">
151      <table class="memname">
152        <tr>
153          <td class="memname">double itpp::besselj           </td>
154          <td>(</td>
155          <td class="paramtype">int&nbsp;</td>
156          <td class="paramname"> <em>nu</em>, </td>
157        </tr>
158        <tr>
159          <td class="paramkey"></td>
160          <td></td>
161          <td class="paramtype">double&nbsp;</td>
162          <td class="paramname"> <em>x</em></td><td>&nbsp;</td>
163        </tr>
164        <tr>
165          <td></td>
166          <td>)</td>
167          <td></td><td></td><td></td>
168        </tr>
169      </table>
170</div>
171<div class="memdoc">
172
173<p>
174Bessel function of first kind of order <em>nu</em> for <em>nu</em> integer.
175<p>
176The bessel function of first kind is defined as: <p class="formulaDsp">
177<img class="formulaDsp" alt="\[ J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k} \]" src="form_171.png">
178<p>
179 where <img class="formulaInl" alt="$\nu$" src="form_172.png"> is the order and <img class="formulaInl" alt="$ 0 < x < \infty $" src="form_173.png">.
180</div>
181</div><p>
182<a class="anchor" name="gff9230fd225656cdf47c384cb7dc37de"></a><!-- doxytag: member="itpp::besselk" ref="gff9230fd225656cdf47c384cb7dc37de" args="(int nu, double x)" -->
183<div class="memitem">
184<div class="memproto">
185      <table class="memname">
186        <tr>
187          <td class="memname">double itpp::besselk           </td>
188          <td>(</td>
189          <td class="paramtype">int&nbsp;</td>
190          <td class="paramname"> <em>nu</em>, </td>
191        </tr>
192        <tr>
193          <td class="paramkey"></td>
194          <td></td>
195          <td class="paramtype">double&nbsp;</td>
196          <td class="paramname"> <em>x</em></td><td>&nbsp;</td>
197        </tr>
198        <tr>
199          <td></td>
200          <td>)</td>
201          <td></td><td></td><td></td>
202        </tr>
203      </table>
204</div>
205<div class="memdoc">
206
207<p>
208Modified Bessel function of second kind of order <em>nu</em>. <em>nu</em> is double. <em>x</em> is double.
209<p>
210The Modified Bessel function of second kind is defined as: <p class="formulaDsp">
211<img class="formulaDsp" alt="\[ K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)] \]" src="form_176.png">
212<p>
213 where <img class="formulaInl" alt="$\nu$" src="form_172.png"> is the order and <img class="formulaInl" alt="$ 0 < x < \infty $" src="form_173.png">.
214</div>
215</div><p>
216<a class="anchor" name="g15832f12203d675afad32e69670130dd"></a><!-- doxytag: member="itpp::bessely" ref="g15832f12203d675afad32e69670130dd" args="(int nu, double x)" -->
217<div class="memitem">
218<div class="memproto">
219      <table class="memname">
220        <tr>
221          <td class="memname">double itpp::bessely           </td>
222          <td>(</td>
223          <td class="paramtype">int&nbsp;</td>
224          <td class="paramname"> <em>nu</em>, </td>
225        </tr>
226        <tr>
227          <td class="paramkey"></td>
228          <td></td>
229          <td class="paramtype">double&nbsp;</td>
230          <td class="paramname"> <em>x</em></td><td>&nbsp;</td>
231        </tr>
232        <tr>
233          <td></td>
234          <td>)</td>
235          <td></td><td></td><td></td>
236        </tr>
237      </table>
238</div>
239<div class="memdoc">
240
241<p>
242Bessel function of second kind of order <em>nu</em>. <em>nu</em> is integer.
243<p>
244The Bessel function of second kind is defined as: <p class="formulaDsp">
245<img class="formulaDsp" alt="\[ Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)} \]" src="form_174.png">
246<p>
247 where <img class="formulaInl" alt="$\nu$" src="form_172.png"> is the order and <img class="formulaInl" alt="$ 0 < x < \infty $" src="form_173.png">.
248</div>
249</div><p>
250</div>
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