Cumulative Distribution Functions (CDF)

Cumulative distribution functions of various distributions.

The functions with the extension _cdf calculate the lower tail integral of the probability density function

\[ D(x) = \int_{-\infty}^{x} p(x') dx' \]

while those with the _cdf_c extension calculate the complement of cumulative distribution function, called in statistics the survival function. It corresponds to the upper tail integral of the probability density function

\[ D(x) = \int_{x}^{+\infty} p(x') dx' \]

NOTE: In the old releases (< 5.14) the _cdf functions were called _quant and the _cdf_c functions were called _prob. These names are currently kept for backward compatibility, but their usage is deprecated.

These functions are defined in the header file Math/ProbFunc.h or in the global one including all statistical functions Math/DistFunc.h

Functions
double	ROOT::Math::beta_cdf (double x, double a, double b)
	Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf. More...
double	ROOT::Math::beta_cdf_c (double x, double a, double b)
	Complement of the cumulative distribution function of the beta distribution. More...
double	ROOT::Math::binomial_cdf (unsigned int k, double p, unsigned int n)
	Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binomial_pdf. More...
double	ROOT::Math::binomial_cdf_c (unsigned int k, double p, unsigned int n)
	Complement of the cumulative distribution function of the Binomial distribution. More...
double	ROOT::Math::breitwigner_cdf (double x, double gamma, double x0=0)
	Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf ) More...
double	ROOT::Math::breitwigner_cdf_c (double x, double gamma, double x0=0)
	Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf_c ) More...
double	ROOT::Math::cauchy_cdf (double x, double b, double x0=0)
	Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian distribution. More...
double	ROOT::Math::cauchy_cdf_c (double x, double b, double x0=0)
	Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is also Lorentzian distribution. More...
double	ROOT::Math::chisquared_cdf (double x, double r, double x0=0)
	Cumulative distribution function of the \(\chi^2\) distribution with \(r\) degrees of freedom (lower tail). More...
double	ROOT::Math::chisquared_cdf_c (double x, double r, double x0=0)
	Complement of the cumulative distribution function of the \(\chi^2\) distribution with \(r\) degrees of freedom (upper tail). More...
double	ROOT::Math::crystalball_cdf (double x, double alpha, double n, double sigma, double x0=0)
	Cumulative distribution for the Crystal Ball distribution function. More...
double	ROOT::Math::crystalball_cdf_c (double x, double alpha, double n, double sigma, double x0=0)
	Complement of the Cumulative distribution for the Crystal Ball distribution. More...
double	ROOT::Math::crystalball_integral (double x, double alpha, double n, double sigma, double x0=0)
	Integral of the not-normalized Crystal Ball function. More...
double	ROOT::Math::exponential_cdf (double x, double lambda, double x0=0)
	Cumulative distribution function of the exponential distribution (lower tail). More...
double	ROOT::Math::exponential_cdf_c (double x, double lambda, double x0=0)
	Complement of the cumulative distribution function of the exponential distribution (upper tail). More...
double	ROOT::Math::fdistribution_cdf (double x, double n, double m, double x0=0)
	Cumulative distribution function of the F-distribution (lower tail). More...
double	ROOT::Math::fdistribution_cdf_c (double x, double n, double m, double x0=0)
	Complement of the cumulative distribution function of the F-distribution (upper tail). More...
double	ROOT::Math::gamma_cdf (double x, double alpha, double theta, double x0=0)
	Cumulative distribution function of the gamma distribution (lower tail). More...
double	ROOT::Math::gamma_cdf_c (double x, double alpha, double theta, double x0=0)
	Complement of the cumulative distribution function of the gamma distribution (upper tail). More...
double	ROOT::Math::landau_cdf (double x, double xi=1, double x0=0)
	Cumulative distribution function of the Landau distribution (lower tail). More...
double	ROOT::Math::landau_cdf_c (double x, double xi=1, double x0=0)
	Complement of the distribution function of the Landau distribution (upper tail). More...
double	ROOT::Math::lognormal_cdf (double x, double m, double s, double x0=0)
	Cumulative distribution function of the lognormal distribution (lower tail). More...
double	ROOT::Math::lognormal_cdf_c (double x, double m, double s, double x0=0)
	Complement of the cumulative distribution function of the lognormal distribution (upper tail). More...
double	ROOT::Math::negative_binomial_cdf (unsigned int k, double p, double n)
	Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of the negative_binomial_pdf. More...
double	ROOT::Math::negative_binomial_cdf_c (unsigned int k, double p, double n)
	Complement of the cumulative distribution function of the Negative Binomial distribution. More...
double	ROOT::Math::normal_cdf (double x, double sigma=1, double x0=0)
	Cumulative distribution function of the normal (Gaussian) distribution (lower tail). More...
double	ROOT::Math::normal_cdf_c (double x, double sigma=1, double x0=0)
	Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail). More...
double	ROOT::Math::poisson_cdf (unsigned int n, double mu)
	Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisson_pdf. More...
double	ROOT::Math::poisson_cdf_c (unsigned int n, double mu)
	Complement of the cumulative distribution function of the Poisson distribution. More...
double	ROOT::Math::tdistribution_cdf (double x, double r, double x0=0)
	Cumulative distribution function of Student's t-distribution (lower tail). More...
double	ROOT::Math::tdistribution_cdf_c (double x, double r, double x0=0)
	Complement of the cumulative distribution function of Student's t-distribution (upper tail). More...
double	ROOT::Math::uniform_cdf (double x, double a, double b, double x0=0)
	Cumulative distribution function of the uniform (flat) distribution (lower tail). More...
double	ROOT::Math::uniform_cdf_c (double x, double a, double b, double x0=0)
	Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail). More...
double	ROOT::Math::vavilov_accurate_cdf (double x, double kappa, double beta2)
	The Vavilov cumulative probability density function. More...
double	ROOT::Math::vavilov_accurate_cdf_c (double x, double kappa, double beta2)
	The Vavilov complementary cumulative probability density function. More...
double	ROOT::Math::vavilov_fast_cdf (double x, double kappa, double beta2)
	The Vavilov cumulative probability density function. More...
double	ROOT::Math::vavilov_fast_cdf_c (double x, double kappa, double beta2)
	The Vavilov complementary cumulative probability density function. More...

Function Documentation

beta_cdf()

double ROOT::Math::beta_cdf	(	double	x,
		double	a,
		double	b
	)

Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.

Definition at line 27 of file ProbFuncMathCore.cxx.

beta_cdf_c()

double ROOT::Math::beta_cdf_c	(	double	x,
		double	a,
		double	b
	)

Complement of the cumulative distribution function of the beta distribution.

Upper tail of the integral of the beta_pdf

Definition at line 20 of file ProbFuncMathCore.cxx.

binomial_cdf()

double ROOT::Math::binomial_cdf	(	unsigned int	k,
		double	p,
		unsigned int	n
	)

Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binomial_pdf.

Definition at line 304 of file ProbFuncMathCore.cxx.

binomial_cdf_c()

double ROOT::Math::binomial_cdf_c	(	unsigned int	k,
		double	p,
		unsigned int	n
	)

Complement of the cumulative distribution function of the Binomial distribution.

Upper tail of the integral of the binomial_pdf

Definition at line 293 of file ProbFuncMathCore.cxx.

breitwigner_cdf()

double ROOT::Math::breitwigner_cdf	(	double	x,
		double	gamma,
		double	x0 = 0
	)

Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf )

\[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{b}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \]

Definition at line 39 of file ProbFuncMathCore.cxx.

breitwigner_cdf_c()

double ROOT::Math::breitwigner_cdf_c	(	double	x,
		double	gamma,
		double	x0 = 0
	)

Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf_c )

\[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \]

Definition at line 33 of file ProbFuncMathCore.cxx.

cauchy_cdf()

double ROOT::Math::cauchy_cdf	(	double	x,
		double	b,
		double	x0 = 0
	)

Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian distribution.

It is similar (just a different parameter definition) to the Breit_Wigner distribution (see breitwigner_cdf )

\[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \]

For detailed description see Mathworld.

Definition at line 51 of file ProbFuncMathCore.cxx.

cauchy_cdf_c()

double ROOT::Math::cauchy_cdf_c	(	double	x,
		double	b,
		double	x0 = 0
	)

Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is also Lorentzian distribution.

It is similar (just a different parameter definition) to the Breit_Wigner distribution (see breitwigner_cdf_c )

\[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \]

For detailed description see Mathworld.

Definition at line 45 of file ProbFuncMathCore.cxx.

chisquared_cdf()

double ROOT::Math::chisquared_cdf	(	double	x,
		double	r,
		double	x0 = 0
	)

Cumulative distribution function of the \(\chi^2\) distribution with \(r\) degrees of freedom (lower tail).

\[ D_{r}(x) = \int_{-\infty}^{x} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c, from Cephes

Definition at line 63 of file ProbFuncMathCore.cxx.

chisquared_cdf_c()

double ROOT::Math::chisquared_cdf_c	(	double	x,
		double	r,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the \(\chi^2\) distribution with \(r\) degrees of freedom (upper tail).

\[ D_{r}(x) = \int_{x}^{+\infty} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c, from Cephes

Definition at line 57 of file ProbFuncMathCore.cxx.

crystalball_cdf()

double ROOT::Math::crystalball_cdf	(	double	x,
		double	alpha,
		double	n,
		double	sigma,
		double	x0 = 0
	)

Cumulative distribution for the Crystal Ball distribution function.

See the definition of the Crystal Ball function at Wikipedia.

The distribution is defined only for n > 1 when the integral converges

Definition at line 69 of file ProbFuncMathCore.cxx.

crystalball_cdf_c()

double ROOT::Math::crystalball_cdf_c	(	double	x,
		double	alpha,
		double	n,
		double	sigma,
		double	x0 = 0
	)

Complement of the Cumulative distribution for the Crystal Ball distribution.

See the definition of the Crystal Ball function at Wikipedia.

The distribution is defined only for n > 1 when the integral converges

Definition at line 84 of file ProbFuncMathCore.cxx.

crystalball_integral()

double ROOT::Math::crystalball_integral	(	double	x,
		double	alpha,
		double	n,
		double	sigma,
		double	x0 = 0
	)

Integral of the not-normalized Crystal Ball function.

See the definition of the Crystal Ball function at Wikipedia.

see ROOT::Math::crystalball_function for the function evaluation.

Definition at line 98 of file ProbFuncMathCore.cxx.

exponential_cdf()

double ROOT::Math::exponential_cdf	(	double	x,
		double	lambda,
		double	x0 = 0
	)

Cumulative distribution function of the exponential distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} \lambda e^{-\lambda x'} dx' \]

For detailed description see Mathworld.

Definition at line 161 of file ProbFuncMathCore.cxx.

exponential_cdf_c()

double ROOT::Math::exponential_cdf_c	(	double	x,
		double	lambda,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the exponential distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} \lambda e^{-\lambda x'} dx' \]

For detailed description see Mathworld.

Definition at line 154 of file ProbFuncMathCore.cxx.

fdistribution_cdf()

double ROOT::Math::fdistribution_cdf	(	double	x,
		double	n,
		double	m,
		double	x0 = 0
	)

Cumulative distribution function of the F-distribution (lower tail).

\[ D_{n,m}(x) = \int_{-\infty}^{x} \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x'^{n/2 -1} (m+nx')^{-(n+m)/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 183 of file ProbFuncMathCore.cxx.

fdistribution_cdf_c()

double ROOT::Math::fdistribution_cdf_c	(	double	x,
		double	n,
		double	m,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the F-distribution (upper tail).

\[ D_{n,m}(x) = \int_{x}^{+\infty} \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x'^{n/2 -1} (m+nx')^{-(n+m)/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 169 of file ProbFuncMathCore.cxx.

gamma_cdf()

double ROOT::Math::gamma_cdf	(	double	x,
		double	alpha,
		double	theta,
		double	x0 = 0
	)

Cumulative distribution function of the gamma distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma, from Cephes

Definition at line 204 of file ProbFuncMathCore.cxx.

gamma_cdf_c()

double ROOT::Math::gamma_cdf_c	(	double	x,
		double	alpha,
		double	theta,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the gamma distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma, from Cephes

Definition at line 198 of file ProbFuncMathCore.cxx.

landau_cdf()

double ROOT::Math::landau_cdf	(	double	x,
		double	xi = 1,
		double	x0 = 0
	)

Cumulative distribution function of the Landau distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} p(x) dx \]

where \(p(x)\) is the Landau probability density function :

\[ p(x) = \frac{1}{\xi} \phi (\lambda) \]

with

\[ \phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds\]

with \(\lambda = (x-x_0)/\xi\). For a detailed description see K.S. Kölbig and B. Schorr, A program package for the Landau distribution, Computer Phys. Comm. 31 (1984) 97-111 [Erratum-ibid. 178 (2008) 972]. The same algorithms as in CERNLIB (DISLAN) is used.

Parameters

x	The argument \(x\)
xi	The width parameter \(\xi\)
x0	The location parameter \(x_0\)

Definition at line 336 of file ProbFuncMathCore.cxx.

landau_cdf_c()

double ROOT::Math::landau_cdf_c ( double  x, double  xi = 1, double  x0 = 0  )

inline

Complement of the distribution function of the Landau distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} p(x) dx \]

where p(x) is the Landau probability density function. It is implemented simply as 1. - landau_cdf

Parameters

x	The argument \(x\)
xi	The width parameter \(\xi\)
x0	The location parameter \(x_0\)

Definition at line 402 of file ProbFuncMathCore.h.

lognormal_cdf()

double ROOT::Math::lognormal_cdf	(	double	x,
		double	m,
		double	s,
		double	x0 = 0
	)

Cumulative distribution function of the lognormal distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \]

For detailed description see Mathworld.

Definition at line 218 of file ProbFuncMathCore.cxx.

lognormal_cdf_c()

double ROOT::Math::lognormal_cdf_c	(	double	x,
		double	m,
		double	s,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the lognormal distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \]

For detailed description see Mathworld.

Definition at line 210 of file ProbFuncMathCore.cxx.

negative_binomial_cdf()

double ROOT::Math::negative_binomial_cdf	(	unsigned int	k,
		double	p,
		double	n
	)

Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of the negative_binomial_pdf.

Definition at line 316 of file ProbFuncMathCore.cxx.

negative_binomial_cdf_c()

double ROOT::Math::negative_binomial_cdf_c	(	unsigned int	k,
		double	p,
		double	n
	)

Complement of the cumulative distribution function of the Negative Binomial distribution.

Upper tail of the integral of the negative_binomial_pdf

Definition at line 326 of file ProbFuncMathCore.cxx.

normal_cdf()

double ROOT::Math::normal_cdf	(	double	x,
		double	sigma = 1,
		double	x0 = 0
	)

Cumulative distribution function of the normal (Gaussian) distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \]

For detailed description see Mathworld.

Definition at line 234 of file ProbFuncMathCore.cxx.

normal_cdf_c()

double ROOT::Math::normal_cdf_c	(	double	x,
		double	sigma = 1,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \]

For detailed description see Mathworld.

Definition at line 226 of file ProbFuncMathCore.cxx.

poisson_cdf()

double ROOT::Math::poisson_cdf	(	unsigned int	n,
		double	mu
	)

Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisson_pdf.

Definition at line 284 of file ProbFuncMathCore.cxx.

poisson_cdf_c()

double ROOT::Math::poisson_cdf_c	(	unsigned int	n,
		double	mu
	)

Complement of the cumulative distribution function of the Poisson distribution.

discrete distributions

Upper tail of the integral of the poisson_pdf

Definition at line 275 of file ProbFuncMathCore.cxx.

tdistribution_cdf()

double ROOT::Math::tdistribution_cdf	(	double	x,
		double	r,
		double	x0 = 0
	)

Cumulative distribution function of Student's t-distribution (lower tail).

\[ D_{r}(x) = \int_{-\infty}^{x} \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x'^2}{r}\right)^{-(r+1)/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 250 of file ProbFuncMathCore.cxx.

tdistribution_cdf_c()

double ROOT::Math::tdistribution_cdf_c	(	double	x,
		double	r,
		double	x0 = 0
	)

Complement of the cumulative distribution function of Student's t-distribution (upper tail).

\[ D_{r}(x) = \int_{x}^{+\infty} \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x'^2}{r}\right)^{-(r+1)/2} dx' \]

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 242 of file ProbFuncMathCore.cxx.

uniform_cdf()

double ROOT::Math::uniform_cdf	(	double	x,
		double	a,
		double	b,
		double	x0 = 0
	)

Cumulative distribution function of the uniform (flat) distribution (lower tail).

\[ D(x) = \int_{-\infty}^{x} {1 \over (b-a)} dx' \]

For detailed description see Mathworld.

Definition at line 266 of file ProbFuncMathCore.cxx.

uniform_cdf_c()

double ROOT::Math::uniform_cdf_c	(	double	x,
		double	a,
		double	b,
		double	x0 = 0
	)

Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail).

\[ D(x) = \int_{x}^{+\infty} {1 \over (b-a)} dx' \]

For detailed description see Mathworld.

Definition at line 258 of file ProbFuncMathCore.cxx.

vavilov_accurate_cdf()

double ROOT::Math::vavilov_accurate_cdf	(	double	x,
		double	kappa,
		double	beta2
	)

The Vavilov cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 472 of file VavilovAccurate.cxx.

vavilov_accurate_cdf_c()

double ROOT::Math::vavilov_accurate_cdf_c	(	double	x,
		double	kappa,
		double	beta2
	)

The Vavilov complementary cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 467 of file VavilovAccurate.cxx.

vavilov_fast_cdf()

double ROOT::Math::vavilov_fast_cdf	(	double	x,
		double	kappa,
		double	beta2
	)

The Vavilov cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(0.01 \le \kappa \le 12 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 582 of file VavilovFast.cxx.

vavilov_fast_cdf_c()

double ROOT::Math::vavilov_fast_cdf_c	(	double	x,
		double	kappa,
		double	beta2
	)

The Vavilov complementary cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(0.01 \le \kappa \le 12 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 587 of file VavilovFast.cxx.