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SpecFuncMathCore.h
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1 // @(#)root/mathcore:$Id$
2 // Authors: Andras Zsenei & Lorenzo Moneta 06/2005
3 
4 /**********************************************************************
5  * *
6  * Copyright (c) 2005 , LCG ROOT MathLib Team *
7  * *
8  * *
9  **********************************************************************/
10 
11 
12 
13 /**
14 
15 Special mathematical functions.
16 The naming and numbering of the functions is taken from
17 <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf">
18 Matt Austern,
19 (Draft) Technical Report on Standard Library Extensions,
20 N1687=04-0127, September 10, 2004</A>
21 
22 @author Created by Andras Zsenei on Mon Nov 8 2004
23 
24 @defgroup SpecFunc Special functions
25 @ingroup MathCore
26 @ingroup MathMore
27 
28 */
29 
30 #ifndef ROOT_Math_SpecFuncMathCore
31 #define ROOT_Math_SpecFuncMathCore
32 
33 
34 namespace ROOT {
35 namespace Math {
36 
37 
38  /** @name Special Functions from MathCore */
39 
40 
41  /**
42 
43  Error function encountered in integrating the normal distribution.
44 
45  \f[ erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \f]
46 
47  For detailed description see
48  <A HREF="http://mathworld.wolfram.com/Erf.html">
49  Mathworld</A>.
50  The implementation used is that of
51  <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC102">GSL</A>.
52  This function is provided only for convenience,
53  in case your standard C++ implementation does not support
54  it. If it does, please use these standard version!
55 
56  @ingroup SpecFunc
57 
58  */
59  // (26.x.21.1) error function
60 
61  double erf(double x);
62 
63 
64 
65  /**
66 
67  Complementary error function.
68 
69  \f[ erfc(x) = 1 - erf(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \f]
70 
71  For detailed description see <A HREF="http://mathworld.wolfram.com/Erfc.html"> Mathworld</A>.
72  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
73 
74  @ingroup SpecFunc
75 
76  */
77  // (26.x.21.2) complementary error function
78 
79  double erfc(double x);
80 
81 
82  /**
83 
84  The gamma function is defined to be the extension of the
85  factorial to real numbers.
86 
87  \f[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \f]
88 
89  For detailed description see
90  <A HREF="http://mathworld.wolfram.com/GammaFunction.html">
91  Mathworld</A>.
92  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
93 
94  @ingroup SpecFunc
95 
96  */
97  // (26.x.18) gamma function
98 
99  double tgamma(double x);
100 
101 
102  /**
103 
104  Calculates the logarithm of the gamma function
105 
106  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
107  @ingroup SpecFunc
108 
109  */
110  double lgamma(double x);
111 
112 
113  /**
114  Calculates the normalized (regularized) lower incomplete gamma function (lower integral)
115 
116  \f[ P(a, x) = \frac{ 1} {\Gamma(a) } \int_{0}^{x} t^{a-1} e^{-t} dt \f]
117 
118 
119  For a detailed description see
120  <A HREF="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
121  Mathworld</A>.
122  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
123  In this implementation both a and x must be positive. If a is negative 1.0 is returned for every x.
124  This is correct only if a is negative integer.
125  For a>0 and x<0 0 is returned (this is correct only for a>0 and x=0).
126 
127  @ingroup SpecFunc
128  */
129  double inc_gamma(double a, double x );
130 
131  /**
132  Calculates the normalized (regularized) upper incomplete gamma function (upper integral)
133 
134  \f[ Q(a, x) = \frac{ 1} {\Gamma(a) } \int_{x}^{\infty} t^{a-1} e^{-t} dt \f]
135 
136 
137  For a detailed description see
138  <A HREF="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
139  Mathworld</A>.
140  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
141  In this implementation both a and x must be positive. If a is negative, 0 is returned for every x.
142  This is correct only if a is negative integer.
143  For a>0 and x<0 1 is returned (this is correct only for a>0 and x=0).
144 
145  @ingroup SpecFunc
146  */
147  double inc_gamma_c(double a, double x );
148 
149  /**
150 
151  Calculates the beta function.
152 
153  \f[ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} \f]
154 
155  for x>0 and y>0. For detailed description see
156  <A HREF="http://mathworld.wolfram.com/BetaDistribution.html">
157  Mathworld</A>.
158 
159  @ingroup SpecFunc
160 
161  */
162  // [5.2.1.3] beta function
163 
164  double beta(double x, double y);
165 
166 
167  /**
168 
169  Calculates the normalized (regularized) incomplete beta function.
170 
171  \f[ B(x, a, b ) = \frac{ \int_{0}^{x} u^{a-1} (1-u)^{b-1} du } { B(a,b) } \f]
172 
173  for 0<=x<=1, a>0, and b>0. For detailed description see
174  <A HREF="http://mathworld.wolfram.com/RegularizedBetaFunction.html">
175  Mathworld</A>.
176  The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier.
177 
178  @ingroup SpecFunc
179 
180  */
181 
182  double inc_beta( double x, double a, double b);
183 
184 
185 
186 
187  /**
188 
189  Calculates the sine integral.
190 
191  \f[ Si(x) = - \int_{0}^{x} \frac{\sin t}{t} dt \f]
192 
193  For detailed description see
194  <A HREF="http://mathworld.wolfram.com/SineIntegral.html">
195  Mathworld</A>. The implementation used is that of
196  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/c336/top.html">
197  CERNLIB</A>,
198  based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326.
199 
200 
201  @ingroup SpecFunc
202 
203  */
204 
205  double sinint(double x);
206 
207 
208 
209 
210  /**
211 
212  Calculates the real part of the cosine integral Re(Ci).
213 
214  For x<0, the imaginary part is \pi i and has to be added by the user,
215  for x>0 the imaginary part of Ci(x) is 0.
216 
217  \f[ Ci(x) = - \int_{x}^{\infty} \frac{\cos t}{t} dt = \gamma + \ln x + \int_{0}^{x} \frac{\cos t - 1}{t} dt\f]
218 
219  For detailed description see
220  <A HREF="http://mathworld.wolfram.com/CosineIntegral.html">
221  Mathworld</A>. The implementation used is that of
222  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/c336/top.html">
223  CERNLIB</A>,
224  based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326.
225 
226 
227  @ingroup SpecFunc
228 
229  */
230 
231  double cosint(double x);
232 
233 
234 
235 
236 } // namespace Math
237 } // namespace ROOT
238 
239 
240 #endif // ROOT_Math_SpecFuncMathCore
double erf(double x)
Error function encountered in integrating the normal distribution.
double cosint(double x)
Calculates the real part of the cosine integral Re(Ci).
Returns the available number of logical cores.
Definition: StringConv.hxx:21
double sinint(double x)
Calculates the sine integral.
double inc_beta(double x, double a, double b)
Calculates the normalized (regularized) incomplete beta function.
double tgamma(double x)
The gamma function is defined to be the extension of the factorial to real numbers.
double beta(double x, double y)
Calculates the beta function.
double erfc(double x)
Complementary error function.
double inc_gamma_c(double a, double x)
Calculates the normalized (regularized) upper incomplete gamma function (upper integral) ...
Double_t x[n]
Definition: legend1.C:17
auto * a
Definition: textangle.C:12
Double_t y[n]
Definition: legend1.C:17
Namespace for new Math classes and functions.
you should not use this method at all Int_t Int_t Double_t Double_t Double_t Int_t Double_t Double_t Double_t Double_t b
Definition: TRolke.cxx:630
double lgamma(double x)
Calculates the logarithm of the gamma function.
double inc_gamma(double a, double x)
Calculates the normalized (regularized) lower incomplete gamma function (lower integral) ...