1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima 5.47.0 Manual: Bessel Functions</title>
<meta name="description" content="Maxima 5.47.0 Manual: Bessel Functions">
<meta name="keywords" content="Maxima 5.47.0 Manual: Bessel Functions">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link href="maxima_toc.html#Top" rel="start" title="Top">
<link href="maxima_423.html#Function-and-Variable-Index" rel="index" title="Function and Variable Index">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_83.html#Special-Functions" rel="up" title="Special Functions">
<link href="maxima_86.html#Airy-Functions" rel="next" title="Airy Functions">
<link href="maxima_84.html#Introduction-to-Special-Functions" rel="previous" title="Introduction to Special Functions">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
body {color: black; background: white; margin-left: 8%; margin-right: 13%;
font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
font-family: sans-serif}
div.synopsisbox {
border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 1em; padding-bottom: 1em;
padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}
-->
</style>
<link rel="icon" href="figures/favicon.ico">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Bessel-Functions"></a>
<div class="header">
<p>
Next: <a href="maxima_86.html#Airy-Functions" accesskey="n" rel="next">Airy Functions</a>, Previous: <a href="maxima_84.html#Introduction-to-Special-Functions" accesskey="p" rel="previous">Introduction to Special Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Bessel-Functions-1"></a>
<h3 class="section">15.2 Bessel Functions</h3>
<a name="bessel_005fj"></a><a name="Item_003a-Special_002fdeffn_002fbessel_005fj"></a><dl>
<dt><a name="index-bessel_005fj"></a>Function: <strong>bessel_j</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The Bessel function of the first kind of order <em>v</em> and argument <em>z</em>.
See <a href="https://personal.math.ubc.ca/~cbm/aands/page_360.htm">A&S eqn 9.1.10</a> and <a href="https://dlmf.nist.gov/10.2.E2">DLMF 10.2.E2</a>.
</p>
<p><code>bessel_j</code> is defined as
</p>
$$
J_v(z) = \sum_{k=0}^{\infty }{{{\left(-1\right)^{k}\,\left(z\over 2\right)^{v+2\,k}
}\over{k!\,\Gamma\left(v+k+1\right)}}}
$$
<p>although the infinite series is not used for computations.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>bessel_j</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="bessel_005fy"></a><a name="Item_003a-Special_002fdeffn_002fbessel_005fy"></a><dl>
<dt><a name="index-bessel_005fy"></a>Function: <strong>bessel_y</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The Bessel function of the second kind of order <em>v</em> and argument <em>z</em>.
See <a href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">A&S eqn 9.1.2</a> and <a href="https://dlmf.nist.gov/10.2.E3">DLMF 10.2.E3</a>.
</p>
<p><code>bessel_y</code> is defined as
$$
Y_v(z) = {{\cos(\pi v)\, J_v(z) - J_{-v}(z)}\over{\sin{\pi v}}}
$$</p>
<p>when <em>v</em> is not an integer. When <em>v</em> is an integer <em>n</em>,
the limit as <em>v</em> approaches <em>n</em> is taken.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>bessel_y</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="bessel_005fi"></a><a name="Item_003a-Special_002fdeffn_002fbessel_005fi"></a><dl>
<dt><a name="index-bessel_005fi"></a>Function: <strong>bessel_i</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The modified Bessel function of the first kind of order <em>v</em> and argument
<em>z</em>. See <a href="https://personal.math.ubc.ca/~cbm/aands/page_375.htm">A&S eqn 9.6.10</a> and <a href="https://dlmf.nist.gov/10.25.E2">DLMF 10.25.E2</a>.
</p>
<p><code>bessel_i</code> is defined as
$$
I_v(z) = \sum_{k=0}^{\infty } {{1\over{k!\,\Gamma
\left(v+k+1\right)}} {\left(z\over 2\right)^{v+2\,k}}}
$$</p>
<p>although the infinite series is not used for computations.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>bessel_i</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="bessel_005fk"></a><a name="Item_003a-Special_002fdeffn_002fbessel_005fk"></a><dl>
<dt><a name="index-bessel_005fk"></a>Function: <strong>bessel_k</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The modified Bessel function of the second kind of order <em>v</em> and argument
<em>z</em>. See <a href="https://personal.math.ubc.ca/~cbm/aands/page_375.htm">A&S eqn 9.6.2</a> and <a href="https://dlmf.nist.gov/10.27.E4">DLMF 10.27.E4</a>.
</p>
<p><code>bessel_k</code> is defined as
$$
K_v(z) = {{\pi\,\csc \left(\pi\,v\right)\,\left(I_{-v}(z)-I_{v}(z)\right)}\over{2}}
$$</p>
<p>when <em>v</em> is not an integer. If <em>v</em> is an integer <em>n</em>,
then the limit as <em>v</em> approaches <em>n</em> is taken.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>bessel_k</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="hankel_005f1"></a><a name="Item_003a-Special_002fdeffn_002fhankel_005f1"></a><dl>
<dt><a name="index-hankel_005f1"></a>Function: <strong>hankel_1</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The Hankel function of the first kind of order <em>v</em> and argument <em>z</em>.
See <a href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">A&S eqn 9.1.3</a> and <a href="https://dlmf.nist.gov/10.4.E3">DLMF 10.4.E3</a>.
</p>
<p><code>hankel_1</code> is defined as
</p>
$$
H^{(1)}_v(z) = J_v(z) + i Y_v(z)
$$
<p>Maxima evaluates <code>hankel_1</code> numerically for a complex order <em>v</em> and
complex argument <em>z</em> in float precision. The numerical evaluation in
bigfloat precision is not supported.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>hankel_1</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<p>Maxima knows the derivative of <code>hankel_1</code> wrt the argument <em>z</em>.
</p>
<p>Examples:
</p>
<p>Numerical evaluation:
</p>
<div class="example">
<pre class="example">(%i1) hankel_1(1,0.5);
(%o1) 0.24226845767487 - 1.471472392670243 %i
</pre><pre class="example">(%i2) hankel_1(1,0.5+%i);
(%o2) - 0.25582879948621 %i - 0.23957560188301
</pre></div>
<p>Expansion of <code>hankel_1</code> when <code>besselexpand</code> is <code>true</code>:
</p>
<div class="example">
<pre class="example">(%i1) hankel_1(1/2,z),besselexpand:true;
sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o1) ----------------------------------
sqrt(%pi) sqrt(z)
</pre></div>
<p>Derivative of <code>hankel_1</code> wrt the argument <em>z</em>. The derivative wrt the
order <em>v</em> is not supported. Maxima returns a noun form:
</p>
<div class="example">
<pre class="example">(%i1) diff(hankel_1(v,z),z);
hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o1) ---------------------------------------
2
</pre><pre class="example">(%i2) diff(hankel_1(v,z),v);
d
(%o2) -- (hankel_1(v, z))
dv
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="hankel_005f2"></a><a name="Item_003a-Special_002fdeffn_002fhankel_005f2"></a><dl>
<dt><a name="index-hankel_005f2"></a>Function: <strong>hankel_2</strong> <em>(<var>v</var>, <var>z</var>)</em></dt>
<dd>
<p>The Hankel function of the second kind of order <em>v</em> and argument <em>z</em>.
See <a href="https://personal.math.ubc.ca/~cbm/aands/page_358.htm">A&S eqn 9.1.4</a> and <a href="https://dlmf.nist.gov/10.4.E3">DLMF 10.4.E3</a>.
</p>
<p><code>hankel_2</code> is defined as
</p>
$$
H^{(2)}_v(z) = J_v(z) - i Y_v(z)
$$
<p>Maxima evaluates <code>hankel_2</code> numerically for a complex order <em>v</em> and
complex argument <em>z</em> in float precision. The numerical evaluation in
bigfloat precision is not supported.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>hankel_2</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code><a href="#besselexpand">besselexpand</a></code>.
</p>
<p>Maxima knows the derivative of <code>hankel_2</code> wrt the argument <em>z</em>.
</p>
<p>For examples see <code>hankel_1</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="besselexpand"></a><a name="Item_003a-Special_002fdefvr_002fbesselexpand"></a><dl>
<dt><a name="index-besselexpand"></a>Option variable: <strong>besselexpand</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p>Controls expansion of the Bessel, Hankel and Struve functions
when the order is half of
an odd integer. In this case, the functions can be expanded
in terms of other elementary functions. When <code>besselexpand</code> is <code>true</code>,
the Bessel function is expanded.
</p>
<div class="example">
<pre class="example">(%i1) besselexpand: false$
(%i2) bessel_j (3/2, z);
3
(%o2) bessel_j(-, z)
2
(%i3) besselexpand: true$
(%i4) bessel_j (3/2, z);
sin(z) cos(z)
sqrt(2) sqrt(z) (------ - ------)
2 z
z
(%o4) ---------------------------------
sqrt(%pi)
(%i5) bessel_y(3/2,z);
sin(z) cos(z)
sqrt(2) sqrt(z) ((- ------) - ------)
z 2
z
(%o5) -------------------------------------
sqrt(%pi)
(%i6) bessel_i(3/2,z);
cosh(z) sinh(z)
sqrt(2) sqrt(z) (------- - -------)
z 2
z
(%o6) -----------------------------------
sqrt(%pi)
(%i7) bessel_k(3/2,z);
1 - z
sqrt(%pi) (- + 1) %e
z
(%o7) -----------------------
sqrt(2) sqrt(z)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="scaled_005fbessel_005fi"></a><a name="Item_003a-Special_002fdeffn_002fscaled_005fbessel_005fi"></a><dl>
<dt><a name="index-scaled_005fbessel_005fi"></a>Function: <strong>scaled_bessel_i</strong> <em>(<var>v</var>, <var>z</var>) </em></dt>
<dd>
<p>The scaled modified Bessel function of the first kind of order
<em>v</em> and argument <em>z</em>. That is,
</p>
$$
{\rm scaled\_bessel\_i}(v,z) = e^{-|z|} I_v(z).
$$
<p>This function is particularly useful
for calculating
\(I_v(z)\) for large <em>z</em>, which is large.
However, maxima does not otherwise know much about this function. For
symbolic work, it is probably preferable to work with the expression
<code>exp(-abs(z))*bessel_i(v, z)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·</div></dd></dl>
<a name="scaled_005fbessel_005fi0"></a><a name="Item_003a-Special_002fdeffn_002fscaled_005fbessel_005fi0"></a><dl>
<dt><a name="index-scaled_005fbessel_005fi0"></a>Function: <strong>scaled_bessel_i0</strong> <em>(<var>z</var>) </em></dt>
<dd>
<p>Identical to <code>scaled_bessel_i(0,z)</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="scaled_005fbessel_005fi1"></a><a name="Item_003a-Special_002fdeffn_002fscaled_005fbessel_005fi1"></a><dl>
<dt><a name="index-scaled_005fbessel_005fi1"></a>Function: <strong>scaled_bessel_i1</strong> <em>(<var>z</var>)</em></dt>
<dd>
<p>Identical to <code>scaled_bessel_i(1,z)</code>.
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></p></dd></dl>
<a name="g_t_0025s"></a><a name="Item_003a-Special_002fdeffn_002f_0025s"></a><dl>
<dt><a name="index-_0025s"></a>Function: <strong>%s</strong> <em>[<var>u</var>,<var>v</var>] (<var>z</var>) </em></dt>
<dd><p>Lommel’s little
\(s_{\mu,\nu}(z)\) function.
(<a href="https://dlmf.nist.gov/11.9.E3">DLMF 11.9.E3</a>)(G&R 8.570.1).
</p>
<p>This Lommel function is the particular solution of the inhomogeneous
Bessel differential equation:
</p>
$$
{d^2\over dz^2} + {1\over z}{dw\over dz} + \left(1-{\nu^2\over z^2}\right) w = z^{\mu-1}
$$
<p>This can be defined by the series
</p>
$$
s_{\mu,\nu}(z) = z^{\mu+1}\sum_{k=0}^{\infty} (-1)^k {z^{2k}\over a_{k+1}(\mu, \nu)}
$$
<p>where
</p>
$$
a_k(\mu,\nu) = \prod_{m=1}^k \left(\left(\mu + 2m-1\right)^2-\nu^2\right) = 4^k\left(\mu-\nu+1\over 2\right)_k \left(\mu+\nu+1\over 2\right)_k
$$
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fdeffn_002fslommel"></a><dl>
<dt><a name="index-slommel"></a>Function: <strong>slommel</strong> <em>[<var>u</var>,<var>v</var>] (<var>z</var>) </em></dt>
<dd><p>Lommel’s big
\(S_{\mu,\nu}(z)\) function.
(<a href="https://dlmf.nist.gov/11.9.E5">DLMF 11.9.E5</a>)(G&R 8.570.2).
</p>
<p>Lommels big S function is another particular solution of the
inhomogeneous Bessel differential equation
(see <a href="#g_t_0025s">%s</a>) defined for all values
of
\(\mu\) and
\(\nu\), where
</p>
$$
\eqalign{
S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} & \Gamma\left({\mu\over 2} + {\nu\over 2} + {1\over 2}\right) \Gamma\left({\mu\over 2} - {\nu\over 2} + {1\over 2}\right) \cr
& \times \left(\sin\left({(\mu-\nu)\pi\over 2}\right) J_{\nu}(z) - \cos\left({(\mu-\nu)\pi\over 2}\right) Y_{\nu}(z)\right)
}
$$
<p>When
\(\mu\pm \nu\)) is an odd
negative integer, the limit must be used.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fnode_002fAiry-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_86.html#Airy-Functions" accesskey="n" rel="next">Airy Functions</a>, Previous: <a href="maxima_84.html#Introduction-to-Special-Functions" accesskey="p" rel="previous">Introduction to Special Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
