-
Notifications
You must be signed in to change notification settings - Fork 11
Expand file tree
/
Copy pathairy.ocd
More file actions
441 lines (417 loc) · 13.2 KB
/
airy.ocd
File metadata and controls
441 lines (417 loc) · 13.2 KB
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
434
435
436
437
438
439
440
<?xml version="1.0"?>
<CD xmlns="http://www.openmath.org/OpenMathCD">
<CDComment>
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work. Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society. In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such. This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
c) The derived work is distributed under terms that allow the
compilation of derived works, but keep paragraphs a) and b)
intact. The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
If you have questions about this license please contact the OpenMath
society at http://www.openmath.org.
</CDComment>
<CDName>airy</CDName>
<CDURL>http://www.openmath.org/CDs/airy.ocd</CDURL>
<CDReviewDate>2017-12-31</CDReviewDate>
<CDDate>2002-01-19</CDDate>
<CDVersion>1</CDVersion>
<CDRevision>2</CDRevision>
<CDComment>
Author: James Davenport
</CDComment>
<CDStatus>experimental</CDStatus>
<Description>
This content dictionary contains symbols to describe the Airy functions
and associated functions.
</Description>
<CDDefinition>
<Name>Ai</Name>
<Description>
The symbol Ai defines the unary Airy Ai function; as in Abramovitz &
Stegun equation 10.4.1. This is a solution to the equation:
$$w^{\prime\prime}-x*w=0$$
It is linearly independent to the Airy Bi function represented by
the Bi symbol in this Content Dictionary and is specifically
given by:
$$Ai(x)=Ai(0)~f(z)-(-Ai^\prime (0))~g(z)$$
where:
$$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k
\frac{z^{3k}}{(3k)!}$$
and:
$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k
\frac{z^{3k+1}}{(3k+1)!}$$
</Description>
<!--
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMS name="Ai" cd="airy"/>
<OMBIND>
<OMS name="ODEsolution" cd="odesoln1"/>
<OMBVAR>
<OMV name="w"/>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS name="plus" cd="arith1"/>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMV name="w"/>
</OMA>
</OMA>
<OMA>
<OMS name="unary_minus" cd="arith1"/>
<OMA>
<OMS name="times" cd="arith1"/>
<OMV name="w"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="list" cd="list1"/>
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMA>
<OMS name="power" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 1 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
<OMA>
<OMS name="times" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="Gamma" cd="euler"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA>
<OMV name="w"/>
<OMS name="zero" cd="alg1"/>
</OMA>
</OMA>
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMA>
<OMS name="unary_minus" cd="arith1"/>
<OMA>
<OMS name="times" cd="arith1"/>
<OMA>
<OMS name="power" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 1 </OMI>
<OMI> 6 </OMI>
</OMA>
</OMA>
<OMA>
<OMS name="Gamma" cd="euler"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="times" cd="arith1"/>
<OMI> 2 </OMI>
<OMS name="pi" cd="nums1"/>
</OMA>
</OMA>
<OMA>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMV name="w"/>
</OMA>
<OMS name="zero" cd="alg1"/>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMA>
</OMOBJ>
</FMP>
-->
</CDDefinition>
<CDDefinition>
<Name>Bi</Name>
<Description>
The symbol Bi defines the unary Airy Bi function. This is defined in
Abramivitz and Stegun 10.4.1 and is a solution to the equation:
$$w^{\prime\prime}-x*w=0$$
It is linearly independant to the Airy Ai function represented by
the Ai symbol in this Content Dictionary and is specifically
given by:
$$Bi(x)=\sqrt{3}(Bi(0)~f(z)+(-Bi^\prime (0))~g(z))$$
where:
$$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k
\frac{z^{3k}}{(3k)!}$$
and:
$$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k
\frac{z^{3k+1}}{(3k+1)!}$$
</Description>
<!--
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMS name="Ai" cd="airy"/>
<OMBIND>
<OMS name="ODEsolution" cd="odesoln1"/>
<OMBVAR>
<OMV name="w"/>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS name="plus" cd="arith1"/>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMV name="w"/>
</OMA>
</OMA>
<OMA>
<OMS name="unary_minus" cd="arith1"/>
<OMA>
<OMS name="times" cd="arith1"/>
<OMV name="w"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="list" cd="list1"/>
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMA>
<OMS name="power" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 5 </OMI>
<OMI> 6 </OMI>
</OMA>
</OMA>
<OMA>
<OMS name="times" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="Gamma" cd="euler"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA>
<OMV name="w"/>
<OMS name="zero" cd="alg1"/>
</OMA>
</OMA>
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMA>
<OMS name="unary_minus" cd="arith1"/>
<OMA>
<OMS name="times" cd="arith1"/>
<OMA>
<OMS name="power" cd="arith1"/>
<OMI> 3 </OMI>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
<OMA>
<OMS name="Gamma" cd="euler"/>
<OMA>
<OMS name="divide" cd="arith1"/>
<OMI> 2 </OMI>
<OMI> 3 </OMI>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA>
<OMS name="times" cd="arith1"/>
<OMI> 2 </OMI>
<OMS name="pi" cd="nums1"/>
</OMA>
</OMA>
<OMA>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMV name="w"/>
</OMA>
<OMS name="zero" cd="alg1"/>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMA>
</OMOBJ>
</FMP>
-->
</CDDefinition>
<CDDefinition>
<Name>Ai2</Name>
<Description>
The symbol Ai2 takes two arguments, it represents derivatives of
the Airy Ai function. The symbol Ai2(n,z) represents the n'th
derivative of Ai(z).
</Description>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="Ai" cd="airy"/>
<OMV name="z"/>
</OMA>
<OMA>
<OMS name="Ai2" cd="airy"/>
<OMS name="zero" cd="alg1"/>
<OMV name="z"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMBIND>
<OMS name="lambda" cd="fns1"/>
<OMBVAR>
<OMV name="z"/>
</OMBVAR>
<OMA>
<OMS name="Ai2" cd="airy"/>
<OMV name="n"/>
<OMV name="z"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="z"/>
</OMA>
<OMA>
<OMS name="Ai2" cd="airy"/>
<OMA>
<OMS name="plus" cd="arith1"/>
<OMV name="n"/>
<OMS name="one" cd="alg1"/>
</OMA>
<OMV name="z"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name>Bi2</Name>
<Description>
The symbol Bi2 takes two arguments, it represents derivatives of
the Airy Bi function. The symbol Bi2(n,z) represents the n'th
derivative of Bi(z).
</Description>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="Bi" cd="airy"/>
<OMV name="z"/>
</OMA>
<OMA>
<OMS name="Bi2" cd="airy"/>
<OMS name="zero" cd="alg1"/>
<OMV name="z"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMBIND>
<OMS name="lambda" cd="fns1"/>
<OMBVAR>
<OMV name="z"/>
</OMBVAR>
<OMA>
<OMS name="Bi2" cd="airy"/>
<OMV name="n"/>
<OMV name="z"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="z"/>
</OMA>
<OMA>
<OMS name="Bi2" cd="airy"/>
<OMA>
<OMS name="plus" cd="arith1"/>
<OMV name="n"/>
<OMS name="one" cd="alg1"/>
</OMA>
<OMV name="z"/>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
</CD>