// @(#)root/mathcore:$Id$ // Authors: Andras Zsenei & Lorenzo Moneta 06/2005 /********************************************************************** * * * Copyright (c) 2005 , LCG ROOT MathLib Team * * * * * **********************************************************************/ /** Special mathematical functions. The naming and numbering of the functions is taken from Matt Austern, (Draft) Technical Report on Standard Library Extensions, N1687=04-0127, September 10, 2004 @author Created by Andras Zsenei on Mon Nov 8 2004 @defgroup SpecFunc Special functions */ #ifndef ROOT_Math_SpecFuncMathCore #define ROOT_Math_SpecFuncMathCore namespace ROOT { namespace Math { /** @name Special Functions from MathCore */ /** Error function encountered in integrating the normal distribution. \f[ erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt \f] For detailed description see Mathworld. The implementation used is that of GSL. This function is provided only for convenience, in case your standard C++ implementation does not support it. If it does, please use these standard version! @ingroup SpecFunc */ // (26.x.21.1) error function double erf(double x); /** Complementary error function. \f[ erfc(x) = 1 - erf(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} e^{-t^2} dt \f] For detailed description see Mathworld. The implementation used is that of Cephes from S. Moshier. @ingroup SpecFunc */ // (26.x.21.2) complementary error function double erfc(double x); /** The gamma function is defined to be the extension of the factorial to real numbers. \f[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \f] For detailed description see Mathworld. The implementation used is that of Cephes from S. Moshier. @ingroup SpecFunc */ // (26.x.18) gamma function double tgamma(double x); /** Calculates the logarithm of the gamma function The implementation used is that of Cephes from S. Moshier. @ingroup SpecFunc */ double lgamma(double x); /** Calculates the normalized (regularized) lower incomplete gamma function (lower integral) \f[ P(a, x) = \frac{ 1} {\Gamma(a) } \int_{0}^{x} t^{a-1} e^{-t} dt \f] For a detailed description see Mathworld. The implementation used is that of Cephes from S. Moshier. In this implementation both a and x must be positive. If a is negative 1.0 is returned for every x. This is correct only if a is negative integer. For a>0 and x<0 0 is returned (this is correct only for a>0 and x=0). @ingroup SpecFunc */ double inc_gamma(double a, double x ); /** Calculates the normalized (regularized) upper incomplete gamma function (upper integral) \f[ Q(a, x) = \frac{ 1} {\Gamma(a) } \int_{x}^{\infty} t^{a-1} e^{-t} dt \f] For a detailed description see Mathworld. The implementation used is that of Cephes from S. Moshier. In this implementation both a and x must be positive. If a is negative, 0 is returned for every x. This is correct only if a is negative integer. For a>0 and x<0 1 is returned (this is correct only for a>0 and x=0). @ingroup SpecFunc */ double inc_gamma_c(double a, double x ); /** Calculates the beta function. \f[ B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} \f] for x>0 and y>0. For detailed description see Mathworld. @ingroup SpecFunc */ // [5.2.1.3] beta function double beta(double x, double y); /** Calculates the normalized (regularized) incomplete beta function. \f[ B(x, a, b ) = \frac{ \int_{0}^{x} u^{a-1} (1-u)^{b-1} du } { B(a,b) } \f] for 0<=x<=1, a>0, and b>0. For detailed description see Mathworld. The implementation used is that of Cephes from S. Moshier. @ingroup SpecFunc */ double inc_beta( double x, double a, double b); /** Calculates the sine integral. \f[ Si(x) = - \int_{0}^{x} \frac{\sin t}{t} dt \f] For detailed description see Mathworld. The implementation used is that of CERNLIB, based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326. @ingroup SpecFunc */ double sinint(double x); /** Calculates the real part of the cosine integral \Re(Ci). For x<0, the imaginary part is \pi i and has to be added by the user, for x>0 the imaginary part of Ci(x) is 0. \f[ Ci(x) = - \int_{x}^{\infty} \frac{\cos t}{t} dt = \gamma + \ln x + \int_{0}^{x} \frac{\cos t - 1}{t} dt\f] For detailed description see Mathworld. The implementation used is that of CERNLIB, based on Y.L. Luke, The special functions and their approximations, v.II, (Academic Press, New York l969) 325-326. @ingroup SpecFunc */ double cosint(double x); } // namespace Math } // namespace ROOT #endif // ROOT_Math_SpecFuncMathCore