hypergeom1F1: Confluent hypergeometric function

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

Confluent hypergeometric function

Description

hypergeom1F1(a, b, z) computes the confluent hypergeometric function 1F1(a, b, z) also known as the Kummer's function M(a, b, z), for the parameters a and b (here assumed to be real scalars), and the complex argument z (scalar, vector or array).

Usage

hypergeom1F1(a, b, z, n)

Arguments

a	parameter a (real scalar).
b	parameter b (real scalar).
z	complex argument (scalar, vector or array).
n	number of the Gauss-Laguerre nodes and weights on (0,Inf). If empty, default value is n = 64.

Details

The present algorithm was adapted from the MATLAB version of the FORTRAN program suggested by S.Zhang and J.Jin (1996). For more details see also http://iris-lee3.ece.uiuc.edu/~jjin/routines/routines.html.

Computation of 1F1(a, b, z) for large arguments z (with |Im(z)| >= |Re(z)|) and for b > a > 0) is based on using the steepest descent integration method as suggested by G. Navas Palencia and A.A. Arratia Quesada (2016).

Value

List including the following items:

f	calculated 1F1(a,b,z) of the same dimension as z,
method	indicator of the used method: -1 is the method used for special cases, 0 is the method used for z == 0, 1 is the method based on the series expansion, 2 is the method based on the steepest descent integration, 3 is the method based on the asymptotic expansion,
loops	indicator of the used number of recursive loops.

Note

Ver.: 23-Sep-2018 17:54:17 (consistent with Matlab CharFunTool v1.3.0, 30-Mar-2018 13:45:00).

References

[1] Jin, J. M., and Zhang Shan Jjie. Computation of Special Functions. Wiley, 1996.

[2] Navas Palencia, G. and Arratia Quesada, A.A., 2016. On the computation of confluent hypergeometric functions for large imaginary part of parameters b and z. In Mathematical Software - ICMS 2016: 5th International Conference, Berlin, Germany, July 11-14, 2016: proceedings, pp. 241-248. Springer.

See Also

For more details on confluent hypergeometric function 1F1(a, b, z) or the Kummer's (confluent hypergeometric) function M(a, b, z) see WIKIPEDIA: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function.

Other Utility Function: ChebCoefficients(), ChebPoints(), ChebPolyValues(), ChebPoly(), ChebValues(), GammaLog(), GammaMultiLog(), GammaMulti(), GammaZX(), Hypergeom1F1MatApprox(), Hypergeom1F1Mat(), Hypergeom2F1Mat(), Hypergeom2F1(), HypergeompFqMat(), InterpChebValues(), interpBarycentric()

Examples

## EXAMPLE 1
a <- 10
b <- 15
z <- c(0, 1, 10i, 50i, 10 + 50i, 100 + 50i)
n <- 64
result <- hypergeom1F1(a, b, z, n)

## EXAMPLE 2
# CF of Beta(1/2,1/2) distribution
a  = 1 / 2
b  = 1 / 2
t  = seq(-100, 100, length.out = 1001)
plotReIm(
        function(t)
                hypergeom1F1(a, a + b, 1i * t)$f,
        t,
        title = "Characteristic function of Beta(1/2,1/2) distribution",
        xlab = "t",
        ylab = "CF"
)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.