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0001 // @(#)root/mathcore:$Id$ 0002 // Authors: Andras Zsenei & Lorenzo Moneta 06/2005 0003 0004 /********************************************************************** 0005 * * 0006 * Copyright (c) 2005 , LCG ROOT MathLib Team * 0007 * * 0008 * * 0009 **********************************************************************/ 0010 0011 /** 0012 0013 Special mathematical functions. 0014 The naming and numbering of the functions is taken from 0015 <A HREF="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1687.pdf"> 0016 Matt Austern, 0017 (Draft) Technical Report on Standard Library Extensions, 0018 N1687=04-0127, September 10, 2004</A> 0019 0020 @author Created by Andras Zsenei on Mon Nov 8 2004 0021 0022 @defgroup SpecFunc Special functions 0023 @ingroup MathCore 0024 0025 */ 0026 0027 #ifndef ROOT_Math_SpecFuncMathCore 0028 #define ROOT_Math_SpecFuncMathCore 0029 0030 0031 namespace ROOT { 0032 namespace Math { 0033 0034 0035 /** @name Special Functions from MathCore */ 0036 0037 0038 /** 0039 0040 Error function encountered in integrating the normal distribution. 0041 0042 \f[ erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \f] 0043 0044 For detailed description see 0045 <A HREF="http://mathworld.wolfram.com/Erf.html"> 0046 Mathworld</A>. 0047 The implementation used is that of 0048 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_7.html#SEC102">GSL</A>. 0049 This function is provided only for convenience, 0050 in case your standard C++ implementation does not support 0051 it. If it does, please use these standard version! 0052 0053 @ingroup SpecFunc 0054 0055 */ 0056 // (26.x.21.1) error function 0057 0058 double erf(double x); 0059 0060 0061 0062 /** 0063 0064 Complementary error function. 0065 0066 \f[ erfc(x) = 1 - erf(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \f] 0067 0068 For detailed description see <A HREF="http://mathworld.wolfram.com/Erfc.html"> Mathworld</A>. 0069 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0070 0071 @ingroup SpecFunc 0072 0073 */ 0074 // (26.x.21.2) complementary error function 0075 0076 double erfc(double x); 0077 0078 0079 /** 0080 0081 The gamma function is defined to be the extension of the 0082 factorial to real numbers. 0083 0084 \f[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \f] 0085 0086 For detailed description see 0087 <A HREF="http://mathworld.wolfram.com/GammaFunction.html"> 0088 Mathworld</A>. 0089 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0090 0091 @ingroup SpecFunc 0092 0093 */ 0094 // (26.x.18) gamma function 0095 0096 double tgamma(double x); 0097 0098 0099 /** 0100 0101 Calculates the logarithm of the gamma function 0102 0103 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0104 @ingroup SpecFunc 0105 0106 */ 0107 double lgamma(double x); 0108 0109 0110 /** 0111 Calculates the normalized (regularized) lower incomplete gamma function (lower integral) 0112 0113 \f[ P(a, x) = \frac{ 1} {\Gamma(a) } \int_{0}^{x} t^{a-1} e^{-t} dt \f] 0114 0115 0116 For a detailed description see 0117 <A HREF="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 0118 Mathworld</A>. 0119 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0120 In this implementation both a and x must be positive. If a is negative 1.0 is returned for every x. 0121 This is correct only if a is negative integer. 0122 For a>0 and x<0 0 is returned (this is correct only for a>0 and x=0). 0123 0124 @ingroup SpecFunc 0125 */ 0126 double inc_gamma(double a, double x ); 0127 0128 /** 0129 Calculates the normalized (regularized) upper incomplete gamma function (upper integral) 0130 0131 \f[ Q(a, x) = \frac{ 1} {\Gamma(a) } \int_{x}^{\infty} t^{a-1} e^{-t} dt \f] 0132 0133 0134 For a detailed description see 0135 <A HREF="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 0136 Mathworld</A>. 0137 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0138 In this implementation both a and x must be positive. If a is negative, 0 is returned for every x. 0139 This is correct only if a is negative integer. 0140 For a>0 and x<0 1 is returned (this is correct only for a>0 and x=0). 0141 0142 @ingroup SpecFunc 0143 */ 0144 double inc_gamma_c(double a, double x ); 0145 0146 /** 0147 0148 Calculates the beta function. 0149 0150 \f[ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} \f] 0151 0152 for x>0 and y>0. For detailed description see 0153 <A HREF="http://mathworld.wolfram.com/BetaDistribution.html"> 0154 Mathworld</A>. 0155 0156 @ingroup SpecFunc 0157 0158 */ 0159 // [5.2.1.3] beta function 0160 0161 double beta(double x, double y); 0162 0163 0164 /** 0165 0166 Calculates the normalized (regularized) incomplete beta function. 0167 0168 \f[ B(x, a, b ) = \frac{ \int_{0}^{x} u^{a-1} (1-u)^{b-1} du } { B(a,b) } \f] 0169 0170 for 0<=x<=1, a>0, and b>0. For detailed description see 0171 <A HREF="http://mathworld.wolfram.com/RegularizedBetaFunction.html"> 0172 Mathworld</A>. 0173 The implementation used is that of <A HREF="http://www.netlib.org/cephes">Cephes</A> from S. Moshier. 0174 0175 @ingroup SpecFunc 0176 0177 */ 0178 0179 double inc_beta( double x, double a, double b); 0180 0181 0182 0183 0184 /** 0185 0186 Calculates the sine integral. 0187 0188 \f[ Si(x) = - \int_{0}^{x} \frac{\sin t}{t} dt \f] 0189 0190 For detailed description see 0191 <A HREF="http://mathworld.wolfram.com/SineIntegral.html"> 0192 Mathworld</A>. The implementation used is that of 0193 <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/c336/top.html"> 0194 CERNLIB</A>, 0195 based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326. 0196 0197 0198 @ingroup SpecFunc 0199 0200 */ 0201 0202 double sinint(double x); 0203 0204 0205 0206 0207 /** 0208 0209 Calculates the real part of the cosine integral Re(Ci). 0210 0211 For x<0, the imaginary part is \f$\pi i\f$ and has to be added by the user, 0212 for x>0 the imaginary part of Ci(x) is 0. 0213 0214 \f[ Ci(x) = - \int_{x}^{\infty} \frac{\cos t}{t} dt = \gamma + \ln x + \int_{0}^{x} \frac{\cos t - 1}{t} dt\f] 0215 0216 For detailed description see 0217 <A HREF="http://mathworld.wolfram.com/CosineIntegral.html"> 0218 Mathworld</A>. The implementation used is that of 0219 <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/c336/top.html"> 0220 CERNLIB</A>, 0221 based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326. 0222 0223 0224 @ingroup SpecFunc 0225 0226 */ 0227 0228 double cosint(double x); 0229 0230 0231 0232 0233 } // namespace Math 0234 } // namespace ROOT 0235 0236 0237 #endif // ROOT_Math_SpecFuncMathCore
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