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Reference Guide
SpecFuncMathMore.h
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1 // @(#)root/mathmore:$Id$
2 // Authors: L. Moneta, A. Zsenei 08/2005
3 
4 // Authors: Andras Zsenei & Lorenzo Moneta 06/2005
5 
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26 
27 /**
28 
29 Special mathematical functions.
30 The naming and numbering of the functions is taken from
31 <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf">
32 Matt Austern,
33 (Draft) Technical Report on Standard Library Extensions,
34 N1687=04-0127, September 10, 2004</A>
35 
36 @author Created by Andras Zsenei on Mon Nov 8 2004
37 
38 @defgroup SpecFunc Special functions
39 
40 */
41 
42 
43 
44 
45 
46 #ifndef ROOT_Math_SpecFuncMathMore
47 #define ROOT_Math_SpecFuncMathMore
48 
49 
50 
51 
52 namespace ROOT {
53 namespace Math {
54 
55  /** @name Special Functions from MathMore */
56 
57 
58  /**
59 
60 
61  Computes the generalized Laguerre polynomials for
62  \f$ n \geq 0, m > -1 \f$.
63  They are defined in terms of the confluent hypergeometric function.
64  For integer values of m they can be defined in terms of the Laguerre polynomials \f$L_n(x)\f$:
65 
66  \f[ L_{n}^{m}(x) = (-1)^{m} \frac{d^m}{dx^m} L_{n+m}(x) \f]
67 
68 
69  For detailed description see
70  <A HREF="http://mathworld.wolfram.com/LaguerrePolynomial.html">Mathworld</A>.
71  The implementation used is that of
72  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Laguerre-Functions.html">GSL</A>.
73 
74  This function is an extension of C++0x, also consistent in GSL,
75  Abramowitz and Stegun 1972 and MatheMathica that uses non-integer values for m.
76  C++0x calls for 'int m', more restrictive than necessary.
77  The definition for was incorrect in 'n1687.pdf', but fixed in
78  <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf">n1836.pdf</A>,
79  the most recent draft as of 2007-07-01
80 
81 
82  @ingroup SpecFunc
83 
84  */
85  // [5.2.1.1] associated Laguerre polynomials
86 
87  double assoc_laguerre(unsigned n, double m, double x);
88 
89 
90 
91 
92  /**
93 
94  Computes the associated Legendre polynomials.
95 
96  \f[ P_{l}^{m}(x) = (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{l}(x) \f]
97 
98  with \f$m \geq 0\f$, \f$ l \geq m \f$ and \f$ |x|<1 \f$.
99  There are two sign conventions for associated Legendre polynomials.
100  As is the case with the above formula, some authors (e.g., Arfken
101  1985, pp. 668-669) omit the Condon-Shortley phase \f$(-1)^m\f$,
102  while others include it (e.g., Abramowitz and Stegun 1972).
103  One possible way to distinguish the two conventions is due to
104  Abramowitz and Stegun (1972, p. 332), who use the notation
105 
106  \f[ P_{lm} (x) = (-1)^m P_{l}^{m} (x)\f]
107 
108  to distinguish the two. For detailed description see
109  <A HREF="http://mathworld.wolfram.com/LegendrePolynomial.html">
110  Mathworld</A>. The implementation used is that of
111  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Associated-Legendre-Polynomials-and-Spherical-Harmonics.html">GSL</A>.
112 
113  The definition uses is the one of C++0x, \f$ P_{lm}\f$, while GSL and MatheMatica use the \f$P_{l}^{m}\f$ definition. Note that C++0x and GSL definitions agree instead for the normalized associated Legendre polynomial,
114  sph_legendre(l,m,theta).
115 
116  @ingroup SpecFunc
117 
118  */
119  // [5.2.1.2] associated Legendre functions
120 
121  double assoc_legendre(unsigned l, unsigned m, double x);
122 
123 
124 
125 
126 
127  /**
128 
129  Calculates the complete elliptic integral of the first kind.
130 
131  \f[ K(k) = F(k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{\sqrt{1 - k^2 \sin^2{\theta}}} \f]
132 
133  with \f$0 \leq k^2 \leq 1\f$. For detailed description see
134  <A HREF="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">
135  Mathworld</A>. The implementation used is that of
136  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC100">GSL</A>.
137 
138  @ingroup SpecFunc
139 
140  */
141  // [5.2.1.4] (complete) elliptic integral of the first kind
142 
143  double comp_ellint_1(double k);
144 
145 
146 
147 
148  /**
149 
150  Calculates the complete elliptic integral of the second kind.
151 
152  \f[ E(k) = E(k , \pi / 2) = \int_{0}^{\pi /2} \sqrt{1 - k^2 \sin^2{\theta}} d \theta \f]
153 
154  with \f$0 \leq k^2 \leq 1\f$. For detailed description see
155  <A HREF="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html">
156  Mathworld</A>. The implementation used is that of
157  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC100">GSL</A>.
158 
159  @ingroup SpecFunc
160 
161  */
162  // [5.2.1.5] (complete) elliptic integral of the second kind
163 
164  double comp_ellint_2(double k);
165 
166 
167 
168 
169  /**
170 
171  Calculates the complete elliptic integral of the third kind.
172 
173  \f[ \Pi (n, k, \pi / 2) = \int_{0}^{\pi /2} \frac{d \theta}{(1 - n \sin^2{\theta})\sqrt{1 - k^2 \sin^2{\theta}}} \f]
174 
175  with \f$0 \leq k^2 \leq 1\f$. There are two sign conventions
176  for elliptic integrals of the third kind. Some authors (Abramowitz
177  and Stegun,
178  <A HREF="http://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html">
179  Mathworld</A>,
180  <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf">
181  C++ standard proposal</A>) use the above formula, while others
182  (<A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC95">
183  GSL</A>, <A HREF="http://planetmath.org/encyclopedia/EllipticIntegralsAndJacobiEllipticFunctions.html">
184  Planetmath</A> and
185  <A HREF="http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/c346/top.html">
186  CERNLIB</A>) use the + sign in front of n in the denominator. In
187  order to be C++ compliant, the present library uses the former
188  convention. The implementation used is that of
189  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC101">GSL</A>.
190 
191  @ingroup SpecFunc
192 
193  */
194  // [5.2.1.6] (complete) elliptic integral of the third kind
195  double comp_ellint_3(double n, double k);
196 
197 
198 
199 
200  /**
201 
202  Calculates the confluent hypergeometric functions of the first kind.
203 
204  \f[ _{1}F_{1}(a;b;z) = \frac{\Gamma(b)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(b+n)} \frac{z^n}{n!} \f]
205 
206  For detailed description see
207  <A HREF="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html">
208  Mathworld</A>. The implementation used is that of
209  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC125">GSL</A>.
210 
211  @ingroup SpecFunc
212 
213  */
214  // [5.2.1.7] confluent hypergeometric functions
215 
216  double conf_hyperg(double a, double b, double z);
217 
218 
219  /**
220 
221  Calculates the confluent hypergeometric functions of the second kind, known also as Kummer function of the second kind,
222  it is related to the confluent hypergeometric functions of the first kind.
223 
224  \f[ U(a,b,z) = \frac{ \pi}{ \sin{\pi b } } \left[ \frac{ _{1}F_{1}(a,b,z) } {\Gamma(a-b+1) }
225  - \frac{ z^{1-b} { _{1}F_{1}}(a-b+1,2-b,z)}{\Gamma(a)} \right] \f]
226 
227  For detailed description see
228  <A HREF="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html">
229  Mathworld</A>. The implementation used is that of
230  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC125">GSL</A>.
231  This function is not part of the C++ standard proposal
232 
233  @ingroup SpecFunc
234 
235  */
236  // confluent hypergeometric functions of second type
237 
238  double conf_hypergU(double a, double b, double z);
239 
240 
241 
242  /**
243 
244  Calculates the modified Bessel function of the first kind
245  (also called regular modified (cylindrical) Bessel function).
246 
247  \f[ I_{\nu} (x) = i^{-\nu} J_{\nu}(ix) = \sum_{k=0}^{\infty} \frac{(\frac{1}{2}x)^{\nu + 2k}}{k! \Gamma(\nu + k + 1)} \f]
248 
249  for \f$x>0, \nu > 0\f$. For detailed description see
250  <A HREF="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">
251  Mathworld</A>. The implementation used is that of
252  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC71">GSL</A>.
253 
254  @ingroup SpecFunc
255 
256  */
257  // [5.2.1.8] regular modified cylindrical Bessel functions
258 
259  double cyl_bessel_i(double nu, double x);
260 
261 
262 
263 
264  /**
265 
266  Calculates the (cylindrical) Bessel functions of the first kind (also
267  called regular (cylindrical) Bessel functions).
268 
269  \f[ J_{\nu} (x) = \sum_{k=0}^{\infty} \frac{(-1)^k(\frac{1}{2}x)^{\nu + 2k}}{k! \Gamma(\nu + k + 1)} \f]
270 
271  For detailed description see
272  <A HREF="http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html">
273  Mathworld</A>. The implementation used is that of
274  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC69">GSL</A>.
275 
276  @ingroup SpecFunc
277 
278  */
279  // [5.2.1.9] cylindrical Bessel functions (of the first kind)
280 
281  double cyl_bessel_j(double nu, double x);
282 
283 
284 
285 
286 
287  /**
288 
289  Calculates the modified Bessel functions of the second kind
290  (also called irregular modified (cylindrical) Bessel functions).
291 
292  \f[ K_{\nu} (x) = \frac{\pi}{2} i^{\nu + 1} (J_{\nu} (ix) + iN(ix)) = \left\{ \begin{array}{cl} \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin{\nu \pi}} & \mbox{for non-integral $\nu$} \\ \frac{\pi}{2} \lim{\mu \to \nu} \frac{I_{-\mu}(x) - I_{\mu}(x)}{\sin{\mu \pi}}
293 & \mbox{for integral $\nu$} \end{array} \right. \f]
294 
295  for \f$x>0, \nu > 0\f$. For detailed description see
296  <A HREF="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html">
297  Mathworld</A>. The implementation used is that of
298  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC72">GSL</A>.
299 
300  @ingroup SpecFunc
301 
302  */
303  // [5.2.1.10] irregular modified cylindrical Bessel functions
304 
305  double cyl_bessel_k(double nu, double x);
306 
307 
308 
309 
310  /**
311 
312  Calculates the (cylindrical) Bessel functions of the second kind
313  (also called irregular (cylindrical) Bessel functions or
314  (cylindrical) Neumann functions).
315 
316  \f[ N_{\nu} (x) = Y_{\nu} (x) = \left\{ \begin{array}{cl} \frac{J_{\nu} \cos{\nu \pi}-J_{-\nu}(x)}{\sin{\nu \pi}} & \mbox{for non-integral $\nu$} \\ \lim{\mu \to \nu} \frac{J_{\mu} \cos{\mu \pi}-J_{-\mu}(x)}{\sin{\mu \pi}} & \mbox{for integral $\nu$} \end{array} \right. \f]
317 
318  For detailed description see
319  <A HREF="http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html">
320  Mathworld</A>. The implementation used is that of
321  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC70">GSL</A>.
322 
323  @ingroup SpecFunc
324 
325  */
326  // [5.2.1.11] cylindrical Neumann functions;
327  // cylindrical Bessel functions (of the second kind)
328 
329  double cyl_neumann(double nu, double x);
330 
331 
332 
333 
334  /**
335 
336  Calculates the incomplete elliptic integral of the first kind.
337 
338  \f[ F(k, \phi) = \int_{0}^{\phi} \frac{d \theta}{\sqrt{1 - k^2 \sin^2{\theta}}} \f]
339 
340  with \f$0 \leq k^2 \leq 1\f$. For detailed description see
341  <A HREF="http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html">
342  Mathworld</A>. The implementation used is that of
343  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC101">GSL</A>.
344 
345  @param k
346  @param phi angle in radians
347 
348  @ingroup SpecFunc
349 
350  */
351  // [5.2.1.12] (incomplete) elliptic integral of the first kind
352  // phi in radians
353 
354  double ellint_1(double k, double phi);
355 
356 
357 
358 
359  /**
360 
361  Calculates the complete elliptic integral of the second kind.
362 
363  \f[ E(k , \phi) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2{\theta}} d \theta \f]
364 
365  with \f$0 \leq k^2 \leq 1\f$. For detailed description see
366  <A HREF="http://mathworld.wolfram.com/EllipticIntegraloftheSecondKind.html">
367  Mathworld</A>. The implementation used is that of
368  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC101">GSL</A>.
369 
370  @param k
371  @param phi angle in radians
372 
373  @ingroup SpecFunc
374 
375  */
376  // [5.2.1.13] (incomplete) elliptic integral of the second kind
377  // phi in radians
378 
379  double ellint_2(double k, double phi);
380 
381 
382 
383 
384  /**
385 
386  Calculates the complete elliptic integral of the third kind.
387 
388  \f[ \Pi (n, k, \phi) = \int_{0}^{\phi} \frac{d \theta}{(1 - n \sin^2{\theta})\sqrt{1 - k^2 \sin^2{\theta}}} \f]
389 
390  with \f$0 \leq k^2 \leq 1\f$. There are two sign conventions
391  for elliptic integrals of the third kind. Some authors (Abramowitz
392  and Stegun,
393  <A HREF="http://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html">
394  Mathworld</A>,
395  <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf">
396  C++ standard proposal</A>) use the above formula, while others
397  (<A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC95">
398  GSL</A>, <A HREF="http://planetmath.org/encyclopedia/EllipticIntegralsAndJacobiEllipticFunctions.html">
399  Planetmath</A> and
400  <A HREF="http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/c346/top.html">
401  CERNLIB</A>) use the + sign in front of n in the denominator. In
402  order to be C++ compliant, the present library uses the former
403  convention. The implementation used is that of
404  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC101">GSL</A>.
405 
406  @param n
407  @param k
408  @param phi angle in radians
409 
410  @ingroup SpecFunc
411 
412  */
413  // [5.2.1.14] (incomplete) elliptic integral of the third kind
414  // phi in radians
415 
416  double ellint_3(double n, double k, double phi);
417 
418 
419 
420 
421  /**
422 
423  Calculates the exponential integral.
424 
425  \f[ Ei(x) = - \int_{-x}^{\infty} \frac{e^{-t}}{t} dt \f]
426 
427  For detailed description see
428  <A HREF="http://mathworld.wolfram.com/ExponentialIntegral.html">
429  Mathworld</A>. The implementation used is that of
430  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC115">GSL</A>.
431 
432  @ingroup SpecFunc
433 
434  */
435  // [5.2.1.15] exponential integral
436 
437  double expint(double x);
438 
439 
440 
441  // [5.2.1.16] Hermite polynomials
442 
443  //double hermite(unsigned n, double x);
444 
445 
446 
447 
448 
449  /**
450 
451  Calculates Gauss' hypergeometric function.
452 
453  \f[ _{2}F_{1}(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a) \Gamma(b)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)} \frac{x^n}{n!} \f]
454 
455  For detailed description see
456  <A HREF="http://mathworld.wolfram.com/HypergeometricFunction.html">
457  Mathworld</A>. The implementation used is that of
458  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC125">GSL</A>.
459 
460  @ingroup SpecFunc
461 
462  */
463  // [5.2.1.17] hypergeometric functions
464 
465  double hyperg(double a, double b, double c, double x);
466 
467 
468 
469  /**
470 
471  Calculates the Laguerre polynomials
472 
473  \f[ P_{l}(x) = \frac{ e^x}{n!} \frac{d^n}{dx^n} (x^n - e^{-x}) \f]
474 
475  for \f$x \geq 0 \f$ in the Rodrigues representation.
476  They corresponds to the associated Laguerre polynomial of order m=0.
477  See Abramowitz and Stegun, (22.5.16)
478  For detailed description see
479  <A HREF="http://mathworld.wolfram.com/LaguerrePolynomial.html">
480  Mathworld</A>.
481  The are implemented using the associated Laguerre polynomial of order m=0.
482 
483  @ingroup SpecFunc
484 
485  */
486  // [5.2.1.18] Laguerre polynomials
487 
488  double laguerre(unsigned n, double x);
489 
490 
491  /**
492 
493  Calculates the Legendre polynomials.
494 
495  \f[ P_{l}(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l \f]
496 
497  for \f$l \geq 0, |x|\leq1\f$ in the Rodrigues representation.
498  For detailed description see
499  <A HREF="http://mathworld.wolfram.com/LegendrePolynomial.html">
500  Mathworld</A>. The implementation used is that of
501  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC129">GSL</A>.
502 
503  @ingroup SpecFunc
504 
505  */
506  // [5.2.1.19] Legendre polynomials
507 
508  double legendre(unsigned l, double x);
509 
510 
511 
512 
513  /**
514 
515  Calculates the Riemann zeta function.
516 
517  \f[ \zeta (x) = \left\{ \begin{array}{cl} \sum_{k=1}^{\infty}k^{-x} & \mbox{for $x > 1$} \\ 2^x \pi^{x-1} \sin{(\frac{1}{2}\pi x)} \Gamma(1-x) \zeta (1-x) & \mbox{for $x < 1$} \end{array} \right. \f]
518 
519  For detailed description see
520  <A HREF="http://mathworld.wolfram.com/RiemannZetaFunction.html">
521  Mathworld</A>. The implementation used is that of
522  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC149">GSL</A>.
523 
524  CHECK WHETHER THE IMPLEMENTATION CALCULATES X<1
525 
526  @ingroup SpecFunc
527 
528  */
529  // [5.2.1.20] Riemann zeta function
530 
531  double riemann_zeta(double x);
532 
533 
534  /**
535 
536  Calculates the spherical Bessel functions of the first kind
537  (also called regular spherical Bessel functions).
538 
539  \f[ j_{n}(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x) \f]
540 
541  For detailed description see
542  <A HREF="http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html">
543  Mathworld</A>. The implementation used is that of
544  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC73">GSL</A>.
545 
546  @ingroup SpecFunc
547 
548  */
549  // [5.2.1.21] spherical Bessel functions of the first kind
550 
551  double sph_bessel(unsigned n, double x);
552 
553 
554  /**
555 
556  Computes the spherical (normalized) associated Legendre polynomials,
557  or spherical harmonic without azimuthal dependence (\f$e^(im\phi)\f$).
558 
559  \f[ Y_l^m(theta,0) = \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(cos \theta) \f]
560 
561  for \f$m \geq 0, l \geq m\f$,
562  where the Condon-Shortley phase \f$(-1)^m\f$ is included in P_l^m(x)
563  This function is consistent with both C++0x and GSL,
564  even though there is a discrepancy in where to include the phase.
565  There is no reference in Abramowitz and Stegun.
566 
567 
568  @ingroup SpecFunc
569 
570  */
571 
572  // [5.2.1.22] spherical associated Legendre functions
573 
574  double sph_legendre(unsigned l, unsigned m, double theta);
575 
576 
577  /**
578 
579  Calculates the spherical Bessel functions of the second kind
580  (also called irregular spherical Bessel functions or
581  spherical Neumann functions).
582 
583  \f[ n_n(x) = y_n(x) = \sqrt{\frac{\pi}{2x}} N_{n+1/2}(x) \f]
584 
585  For detailed description see
586  <A HREF="http://mathworld.wolfram.com/SphericalBesselFunctionoftheSecondKind.html">
587  Mathworld</A>. The implementation used is that of
588  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC74">GSL</A>.
589 
590  @ingroup SpecFunc
591 
592  */
593  // [5.2.1.23] spherical Neumann functions
594 
595  double sph_neumann(unsigned n, double x);
596 
597  /**
598 
599  Calculates the Airy function Ai
600 
601  \f[ Ai(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} \cos(xt + t^3/3) dt \f]
602 
603  For detailed description see
604  <A HREF="http://mathworld.wolfram.com/AiryFunctions.html">
605  Mathworld</A>
606  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
607  The implementation used is that of
608  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Airy-Functions.html">GSL</A>.
609 
610  @ingroup SpecFunc
611 
612  */
613  // Airy function Ai
614 
615  double airy_Ai(double x);
616 
617  /**
618 
619  Calculates the Airy function Bi
620 
621  \f[ Bi(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} [\exp(xt-t^3/3) + \cos(xt + t^3/3)] dt \f]
622 
623  For detailed description see
624  <A HREF="http://mathworld.wolfram.com/AiryFunctions.html">
625  Mathworld</A>
626  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
627  The implementation used is that of
628  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Airy-Functions.html">GSL</A>.
629 
630  @ingroup SpecFunc
631 
632  */
633  // Airy function Bi
634 
635  double airy_Bi(double x);
636 
637  /**
638 
639  Calculates the derivative of the Airy function Ai
640 
641  \f[ Ai(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} \cos(xt + t^3/3) dt \f]
642 
643  For detailed description see
644  <A HREF="http://mathworld.wolfram.com/AiryFunctions.html">
645  Mathworld</A>
646  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
647  The implementation used is that of
648  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Derivatives-of-Airy-Functions.html">GSL</A>.
649 
650  @ingroup SpecFunc
651 
652  */
653  // Derivative of the Airy function Ai
654 
655  double airy_Ai_deriv(double x);
656 
657  /**
658 
659  Calculates the derivative of the Airy function Bi
660 
661  \f[ Bi(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} [\exp(xt-t^3/3) + \cos(xt + t^3/3)] dt \f]
662 
663  For detailed description see
664  <A HREF="http://mathworld.wolfram.com/AiryFunctions.html">
665  Mathworld</A>
666  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
667  The implementation used is that of
668  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Derivatives-of-Airy-Functions.html">GSL</A>.
669 
670  @ingroup SpecFunc
671 
672  */
673  // Derivative of the Airy function Bi
674 
675  double airy_Bi_deriv(double x);
676 
677  /**
678 
679  Calculates the zeroes of the Airy function Ai
680 
681  \f[ Ai(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} \cos(xt + t^3/3) dt \f]
682 
683  For detailed description see
684  <A HREF="http://mathworld.wolfram.com/AiryFunctionZeros.html">
685  Mathworld</A>
686  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
687  The implementation used is that of
688  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Zeros-of-Airy-Functions.html">GSL</A>.
689 
690  @ingroup SpecFunc
691 
692  */
693  // s-th zero of the Airy function Ai
694 
695  double airy_zero_Ai(unsigned int s);
696 
697  /**
698 
699  Calculates the zeroes of the Airy function Bi
700 
701  \f[ Bi(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} [\exp(xt-t^3/3) + \cos(xt + t^3/3)] dt \f]
702 
703  For detailed description see
704  <A HREF="http://mathworld.wolfram.com/AiryFunctionZeros.html">
705  Mathworld</A>
706  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
707  The implementation used is that of
708  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Zeros-of-Airy-Functions.html">GSL</A>.
709 
710  @ingroup SpecFunc
711 
712  */
713  // s-th zero of the Airy function Bi
714 
715  double airy_zero_Bi(unsigned int s);
716 
717  /**
718 
719  Calculates the zeroes of the derivative of the Airy function Ai
720 
721  \f[ Ai(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} \cos(xt + t^3/3) dt \f]
722 
723  For detailed description see
724  <A HREF="http://mathworld.wolfram.com/AiryFunctionZeros.html">
725  Mathworld</A>
726  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
727  The implementation used is that of
728  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Zeros-of-Derivatives-of-Airy-Functions.html">GSL</A>.
729 
730  @ingroup SpecFunc
731 
732  */
733  // s-th zero of the derivative of the Airy function Ai
734 
735  double airy_zero_Ai_deriv(unsigned int s);
736 
737  /**
738 
739  Calculates the zeroes of the derivative of the Airy function Bi
740 
741  \f[ Bi(x) = \frac{1}{\pi} \int\limits_{0}^{\infty} [\exp(xt-t^3/3) + \cos(xt + t^3/3)] dt \f]
742 
743  For detailed description see
744  <A HREF="http://mathworld.wolfram.com/AiryFunctionZeros.html">
745  Mathworld</A>
746  and <A HREF="http://www.nrbook.com/abramowitz_and_stegun/page_446.htm">Abramowitz&Stegun, Sect. 10.4</A>.
747  The implementation used is that of
748  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/Zeros-of-Derivatives-of-Airy-Functions.html">GSL</A>.
749 
750  @ingroup SpecFunc
751 
752  */
753  // s-th zero of the derivative of the Airy function Bi
754 
755  double airy_zero_Bi_deriv(unsigned int s);
756 
757  /**
758 
759  Calculates the Wigner 3j coupling coefficients
760 
761  (ja jb jc
762  ma mb mc)
763 
764  where ja,ma,...etc are integers or half integers.
765  The function takes as input arguments only integers which corresponds
766  to half integer units, e.g two_ja = 2 * ja
767 
768  For detailed description see
769  <A HREF="http://mathworld.wolfram.com/Wigner3j-Symbol.html.html">
770  Mathworld</A>.
771  The implementation used is that of
772  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/3_002dj-Symbols.html#g_t3_002dj-Symbols">GSL</A>.
773 
774  @ingroup SpecFunc
775 
776  */
777 
778  double wigner_3j(int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc);
779 
780  /**
781 
782  Calculates the Wigner 6j coupling coefficients
783 
784  (ja jb jc
785  jd je jf)
786 
787  where ja,jb,...etc are integers or half integers.
788  The function takes as input arguments only integers which corresponds
789  to half integer units, e.g two_ja = 2 * ja
790 
791  For detailed description see
792  <A HREF="http://mathworld.wolfram.com/Wigner6j-Symbol.html">
793  Mathworld</A>.
794  The implementation used is that of
795  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/6_002dj-Symbols.html#g_t6_002dj-Symbols">GSL</A>.
796 
797  @ingroup SpecFunc
798 
799  */
800 
801  double wigner_6j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf);
802 
803  /**
804 
805  Calculates the Wigner 9j coupling coefficients
806 
807  (ja jb jc
808  jd je jf
809  jg jh ji)
810 
811  where ja,jb...etc are integers or half integers.
812  The function takes as input arguments only integers which corresponds
813  to half integer units, e.g two_ja = 2 * ja
814 
815 
816  For detailed description see
817  <A HREF="http://mathworld.wolfram.com/Wigner9j-Symbol.html">
818  Mathworld</A>.
819  The implementation used is that of
820  <A HREF="http://www.gnu.org/software/gsl/manual/html_node/9_002dj-Symbols.html#g_t9_002dj-Symbols">GSL</A>.
821 
822  @ingroup SpecFunc
823 
824  */
825 
826  double wigner_9j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji);
827 
828 
829 
830 } // namespace Math
831 } // namespace ROOT
832 
833 
834 #endif //ROOT_Math_SpecFuncMathMore
double airy_zero_Bi(unsigned int s)
Calculates the zeroes of the Airy function Bi.
Namespace for new ROOT classes and functions.
Definition: ROOT.py:1
double sph_legendre(unsigned l, unsigned m, double theta)
Computes the spherical (normalized) associated Legendre polynomials, or spherical harmonic without az...
double cyl_bessel_j(double nu, double x)
Calculates the (cylindrical) Bessel functions of the first kind (also called regular (cylindrical) Be...
double airy_zero_Ai_deriv(unsigned int s)
Calculates the zeroes of the derivative of the Airy function Ai.
TArc * a
Definition: textangle.C:12
double ellint_2(double k, double phi)
Calculates the complete elliptic integral of the second kind.
double wigner_6j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf)
Calculates the Wigner 6j coupling coefficients.
double legendre(unsigned l, double x)
Calculates the Legendre polynomials.
double comp_ellint_1(double k)
Calculates the complete elliptic integral of the first kind.
double sph_bessel(unsigned n, double x)
Calculates the spherical Bessel functions of the first kind (also called regular spherical Bessel fun...
Double_t x[n]
Definition: legend1.C:17
double conf_hyperg(double a, double b, double z)
Calculates the confluent hypergeometric functions of the first kind.
double hyperg(double a, double b, double c, double x)
Calculates Gauss' hypergeometric function.
double wigner_9j(int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji)
Calculates the Wigner 9j coupling coefficients.
double airy_zero_Bi_deriv(unsigned int s)
Calculates the zeroes of the derivative of the Airy function Bi.
double expint(double x)
Calculates the exponential integral.
double ellint_3(double n, double k, double phi)
Calculates the complete elliptic integral of the third kind.
double riemann_zeta(double x)
Calculates the Riemann zeta function.
double comp_ellint_2(double k)
Calculates the complete elliptic integral of the second kind.
TMarker * m
Definition: textangle.C:8
double assoc_laguerre(unsigned n, double m, double x)
Computes the generalized Laguerre polynomials for .
double assoc_legendre(unsigned l, unsigned m, double x)
Computes the associated Legendre polynomials.
TLine * l
Definition: textangle.C:4
double airy_zero_Ai(unsigned int s)
Calculates the zeroes of the Airy function Ai.
double airy_Bi(double x)
Calculates the Airy function Bi.
double laguerre(unsigned n, double x)
Calculates the Laguerre polynomials.
double airy_Bi_deriv(double x)
Calculates the derivative of the Airy function Bi.
double comp_ellint_3(double n, double k)
Calculates the complete elliptic integral of the third kind.
double airy_Ai_deriv(double x)
Calculates the derivative of the Airy function Ai.
double ellint_1(double k, double phi)
Calculates the incomplete elliptic integral of the first kind.
double cyl_neumann(double nu, double x)
Calculates the (cylindrical) Bessel functions of the second kind (also called irregular (cylindrical)...
Namespace for new Math classes and functions.
double wigner_3j(int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc)
Calculates the Wigner 3j coupling coefficients.
double sph_neumann(unsigned n, double x)
Calculates the spherical Bessel functions of the second kind (also called irregular spherical Bessel ...
double conf_hypergU(double a, double b, double z)
Calculates the confluent hypergeometric functions of the second kind, known also as Kummer function o...
double cyl_bessel_i(double nu, double x)
Calculates the modified Bessel function of the first kind (also called regular modified (cylindrical)...
double airy_Ai(double x)
Calculates the Airy function Ai.
const Int_t n
Definition: legend1.C:16
double cyl_bessel_k(double nu, double x)
Calculates the modified Bessel functions of the second kind (also called irregular modified (cylindri...