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Diffstat (limited to 'lib/Numeric/GSL/Special/gsl_sf_legendre.h')
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diff --git a/lib/Numeric/GSL/Special/gsl_sf_legendre.h b/lib/Numeric/GSL/Special/gsl_sf_legendre.h new file mode 100644 index 0000000..f8068f4 --- /dev/null +++ b/lib/Numeric/GSL/Special/gsl_sf_legendre.h | |||
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1 | /* specfunc/gsl_sf_legendre.h | ||
2 | * | ||
3 | * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2004 Gerard Jungman | ||
4 | * | ||
5 | * This program is free software; you can redistribute it and/or modify | ||
6 | * it under the terms of the GNU General Public License as published by | ||
7 | * the Free Software Foundation; either version 2 of the License, or (at | ||
8 | * your option) any later version. | ||
9 | * | ||
10 | * This program is distributed in the hope that it will be useful, but | ||
11 | * WITHOUT ANY WARRANTY; without even the implied warranty of | ||
12 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | ||
13 | * General Public License for more details. | ||
14 | * | ||
15 | * You should have received a copy of the GNU General Public License | ||
16 | * along with this program; if not, write to the Free Software | ||
17 | * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. | ||
18 | */ | ||
19 | |||
20 | /* Author: G. Jungman */ | ||
21 | |||
22 | #ifndef __GSL_SF_LEGENDRE_H__ | ||
23 | #define __GSL_SF_LEGENDRE_H__ | ||
24 | |||
25 | #include <gsl/gsl_sf_result.h> | ||
26 | |||
27 | #undef __BEGIN_DECLS | ||
28 | #undef __END_DECLS | ||
29 | #ifdef __cplusplus | ||
30 | # define __BEGIN_DECLS extern "C" { | ||
31 | # define __END_DECLS } | ||
32 | #else | ||
33 | # define __BEGIN_DECLS /* empty */ | ||
34 | # define __END_DECLS /* empty */ | ||
35 | #endif | ||
36 | |||
37 | __BEGIN_DECLS | ||
38 | |||
39 | |||
40 | /* P_l(x) l >= 0; |x| <= 1 | ||
41 | * | ||
42 | * exceptions: GSL_EDOM | ||
43 | */ | ||
44 | int gsl_sf_legendre_Pl_e(const int l, const double x, gsl_sf_result * result); | ||
45 | double gsl_sf_legendre_Pl(const int l, const double x); | ||
46 | |||
47 | |||
48 | /* P_l(x) for l=0,...,lmax; |x| <= 1 | ||
49 | * | ||
50 | * exceptions: GSL_EDOM | ||
51 | */ | ||
52 | int gsl_sf_legendre_Pl_array( | ||
53 | const int lmax, const double x, | ||
54 | double * result_array | ||
55 | ); | ||
56 | |||
57 | |||
58 | /* P_l(x) and P_l'(x) for l=0,...,lmax; |x| <= 1 | ||
59 | * | ||
60 | * exceptions: GSL_EDOM | ||
61 | */ | ||
62 | int gsl_sf_legendre_Pl_deriv_array( | ||
63 | const int lmax, const double x, | ||
64 | double * result_array, | ||
65 | double * result_deriv_array | ||
66 | ); | ||
67 | |||
68 | |||
69 | /* P_l(x), l=1,2,3 | ||
70 | * | ||
71 | * exceptions: none | ||
72 | */ | ||
73 | int gsl_sf_legendre_P1_e(double x, gsl_sf_result * result); | ||
74 | int gsl_sf_legendre_P2_e(double x, gsl_sf_result * result); | ||
75 | int gsl_sf_legendre_P3_e(double x, gsl_sf_result * result); | ||
76 | double gsl_sf_legendre_P1(const double x); | ||
77 | double gsl_sf_legendre_P2(const double x); | ||
78 | double gsl_sf_legendre_P3(const double x); | ||
79 | |||
80 | |||
81 | /* Q_0(x), x > -1, x != 1 | ||
82 | * | ||
83 | * exceptions: GSL_EDOM | ||
84 | */ | ||
85 | int gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result); | ||
86 | double gsl_sf_legendre_Q0(const double x); | ||
87 | |||
88 | |||
89 | /* Q_1(x), x > -1, x != 1 | ||
90 | * | ||
91 | * exceptions: GSL_EDOM | ||
92 | */ | ||
93 | int gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result); | ||
94 | double gsl_sf_legendre_Q1(const double x); | ||
95 | |||
96 | |||
97 | /* Q_l(x), x > -1, x != 1, l >= 0 | ||
98 | * | ||
99 | * exceptions: GSL_EDOM | ||
100 | */ | ||
101 | int gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result); | ||
102 | double gsl_sf_legendre_Ql(const int l, const double x); | ||
103 | |||
104 | |||
105 | /* P_l^m(x) m >= 0; l >= m; |x| <= 1.0 | ||
106 | * | ||
107 | * Note that this function grows combinatorially with l. | ||
108 | * Therefore we can easily generate an overflow for l larger | ||
109 | * than about 150. | ||
110 | * | ||
111 | * There is no trouble for small m, but when m and l are both large, | ||
112 | * then there will be trouble. Rather than allow overflows, these | ||
113 | * functions refuse to calculate when they can sense that l and m are | ||
114 | * too big. | ||
115 | * | ||
116 | * If you really want to calculate a spherical harmonic, then DO NOT | ||
117 | * use this. Instead use legendre_sphPlm() below, which uses a similar | ||
118 | * recursion, but with the normalized functions. | ||
119 | * | ||
120 | * exceptions: GSL_EDOM, GSL_EOVRFLW | ||
121 | */ | ||
122 | int gsl_sf_legendre_Plm_e(const int l, const int m, const double x, gsl_sf_result * result); | ||
123 | double gsl_sf_legendre_Plm(const int l, const int m, const double x); | ||
124 | |||
125 | |||
126 | /* P_l^m(x) m >= 0; l >= m; |x| <= 1.0 | ||
127 | * l=|m|,...,lmax | ||
128 | * | ||
129 | * exceptions: GSL_EDOM, GSL_EOVRFLW | ||
130 | */ | ||
131 | int gsl_sf_legendre_Plm_array( | ||
132 | const int lmax, const int m, const double x, | ||
133 | double * result_array | ||
134 | ); | ||
135 | |||
136 | |||
137 | /* P_l^m(x) and d(P_l^m(x))/dx; m >= 0; lmax >= m; |x| <= 1.0 | ||
138 | * l=|m|,...,lmax | ||
139 | * | ||
140 | * exceptions: GSL_EDOM, GSL_EOVRFLW | ||
141 | */ | ||
142 | int gsl_sf_legendre_Plm_deriv_array( | ||
143 | const int lmax, const int m, const double x, | ||
144 | double * result_array, | ||
145 | double * result_deriv_array | ||
146 | ); | ||
147 | |||
148 | |||
149 | /* P_l^m(x), normalized properly for use in spherical harmonics | ||
150 | * m >= 0; l >= m; |x| <= 1.0 | ||
151 | * | ||
152 | * There is no overflow problem, as there is for the | ||
153 | * standard normalization of P_l^m(x). | ||
154 | * | ||
155 | * Specifically, it returns: | ||
156 | * | ||
157 | * sqrt((2l+1)/(4pi)) sqrt((l-m)!/(l+m)!) P_l^m(x) | ||
158 | * | ||
159 | * exceptions: GSL_EDOM | ||
160 | */ | ||
161 | int gsl_sf_legendre_sphPlm_e(const int l, int m, const double x, gsl_sf_result * result); | ||
162 | double gsl_sf_legendre_sphPlm(const int l, const int m, const double x); | ||
163 | |||
164 | |||
165 | /* sphPlm(l,m,x) values | ||
166 | * m >= 0; l >= m; |x| <= 1.0 | ||
167 | * l=|m|,...,lmax | ||
168 | * | ||
169 | * exceptions: GSL_EDOM | ||
170 | */ | ||
171 | int gsl_sf_legendre_sphPlm_array( | ||
172 | const int lmax, int m, const double x, | ||
173 | double * result_array | ||
174 | ); | ||
175 | |||
176 | |||
177 | /* sphPlm(l,m,x) and d(sphPlm(l,m,x))/dx values | ||
178 | * m >= 0; l >= m; |x| <= 1.0 | ||
179 | * l=|m|,...,lmax | ||
180 | * | ||
181 | * exceptions: GSL_EDOM | ||
182 | */ | ||
183 | int gsl_sf_legendre_sphPlm_deriv_array( | ||
184 | const int lmax, const int m, const double x, | ||
185 | double * result_array, | ||
186 | double * result_deriv_array | ||
187 | ); | ||
188 | |||
189 | |||
190 | |||
191 | /* size of result_array[] needed for the array versions of Plm | ||
192 | * (lmax - m + 1) | ||
193 | */ | ||
194 | int gsl_sf_legendre_array_size(const int lmax, const int m); | ||
195 | |||
196 | |||
197 | /* Irregular Spherical Conical Function | ||
198 | * P^{1/2}_{-1/2 + I lambda}(x) | ||
199 | * | ||
200 | * x > -1.0 | ||
201 | * exceptions: GSL_EDOM | ||
202 | */ | ||
203 | int gsl_sf_conicalP_half_e(const double lambda, const double x, gsl_sf_result * result); | ||
204 | double gsl_sf_conicalP_half(const double lambda, const double x); | ||
205 | |||
206 | |||
207 | /* Regular Spherical Conical Function | ||
208 | * P^{-1/2}_{-1/2 + I lambda}(x) | ||
209 | * | ||
210 | * x > -1.0 | ||
211 | * exceptions: GSL_EDOM | ||
212 | */ | ||
213 | int gsl_sf_conicalP_mhalf_e(const double lambda, const double x, gsl_sf_result * result); | ||
214 | double gsl_sf_conicalP_mhalf(const double lambda, const double x); | ||
215 | |||
216 | |||
217 | /* Conical Function | ||
218 | * P^{0}_{-1/2 + I lambda}(x) | ||
219 | * | ||
220 | * x > -1.0 | ||
221 | * exceptions: GSL_EDOM | ||
222 | */ | ||
223 | int gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result); | ||
224 | double gsl_sf_conicalP_0(const double lambda, const double x); | ||
225 | |||
226 | |||
227 | /* Conical Function | ||
228 | * P^{1}_{-1/2 + I lambda}(x) | ||
229 | * | ||
230 | * x > -1.0 | ||
231 | * exceptions: GSL_EDOM | ||
232 | */ | ||
233 | int gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result); | ||
234 | double gsl_sf_conicalP_1(const double lambda, const double x); | ||
235 | |||
236 | |||
237 | /* Regular Spherical Conical Function | ||
238 | * P^{-1/2-l}_{-1/2 + I lambda}(x) | ||
239 | * | ||
240 | * x > -1.0, l >= -1 | ||
241 | * exceptions: GSL_EDOM | ||
242 | */ | ||
243 | int gsl_sf_conicalP_sph_reg_e(const int l, const double lambda, const double x, gsl_sf_result * result); | ||
244 | double gsl_sf_conicalP_sph_reg(const int l, const double lambda, const double x); | ||
245 | |||
246 | |||
247 | /* Regular Cylindrical Conical Function | ||
248 | * P^{-m}_{-1/2 + I lambda}(x) | ||
249 | * | ||
250 | * x > -1.0, m >= -1 | ||
251 | * exceptions: GSL_EDOM | ||
252 | */ | ||
253 | int gsl_sf_conicalP_cyl_reg_e(const int m, const double lambda, const double x, gsl_sf_result * result); | ||
254 | double gsl_sf_conicalP_cyl_reg(const int m, const double lambda, const double x); | ||
255 | |||
256 | |||
257 | /* The following spherical functions are specializations | ||
258 | * of Legendre functions which give the regular eigenfunctions | ||
259 | * of the Laplacian on a 3-dimensional hyperbolic space. | ||
260 | * Of particular interest is the flat limit, which is | ||
261 | * Flat-Lim := {lambda->Inf, eta->0, lambda*eta fixed}. | ||
262 | */ | ||
263 | |||
264 | /* Zeroth radial eigenfunction of the Laplacian on the | ||
265 | * 3-dimensional hyperbolic space. | ||
266 | * | ||
267 | * legendre_H3d_0(lambda,eta) := sin(lambda*eta)/(lambda*sinh(eta)) | ||
268 | * | ||
269 | * Normalization: | ||
270 | * Flat-Lim legendre_H3d_0(lambda,eta) = j_0(lambda*eta) | ||
271 | * | ||
272 | * eta >= 0.0 | ||
273 | * exceptions: GSL_EDOM | ||
274 | */ | ||
275 | int gsl_sf_legendre_H3d_0_e(const double lambda, const double eta, gsl_sf_result * result); | ||
276 | double gsl_sf_legendre_H3d_0(const double lambda, const double eta); | ||
277 | |||
278 | |||
279 | /* First radial eigenfunction of the Laplacian on the | ||
280 | * 3-dimensional hyperbolic space. | ||
281 | * | ||
282 | * legendre_H3d_1(lambda,eta) := | ||
283 | * 1/sqrt(lambda^2 + 1) sin(lam eta)/(lam sinh(eta)) | ||
284 | * (coth(eta) - lambda cot(lambda*eta)) | ||
285 | * | ||
286 | * Normalization: | ||
287 | * Flat-Lim legendre_H3d_1(lambda,eta) = j_1(lambda*eta) | ||
288 | * | ||
289 | * eta >= 0.0 | ||
290 | * exceptions: GSL_EDOM | ||
291 | */ | ||
292 | int gsl_sf_legendre_H3d_1_e(const double lambda, const double eta, gsl_sf_result * result); | ||
293 | double gsl_sf_legendre_H3d_1(const double lambda, const double eta); | ||
294 | |||
295 | |||
296 | /* l'th radial eigenfunction of the Laplacian on the | ||
297 | * 3-dimensional hyperbolic space. | ||
298 | * | ||
299 | * Normalization: | ||
300 | * Flat-Lim legendre_H3d_l(l,lambda,eta) = j_l(lambda*eta) | ||
301 | * | ||
302 | * eta >= 0.0, l >= 0 | ||
303 | * exceptions: GSL_EDOM | ||
304 | */ | ||
305 | int gsl_sf_legendre_H3d_e(const int l, const double lambda, const double eta, gsl_sf_result * result); | ||
306 | double gsl_sf_legendre_H3d(const int l, const double lambda, const double eta); | ||
307 | |||
308 | |||
309 | /* Array of H3d(ell), 0 <= ell <= lmax | ||
310 | */ | ||
311 | int gsl_sf_legendre_H3d_array(const int lmax, const double lambda, const double eta, double * result_array); | ||
312 | |||
313 | |||
314 | |||
315 | |||
316 | |||
317 | __END_DECLS | ||
318 | |||
319 | #endif /* __GSL_SF_LEGENDRE_H__ */ | ||