besselK_inc_clo: Exact Calculation of the Incomplete Bessel K Function

Description

Calculates upper and lower incomplete functions of the modified Bessel function of the third kind $K_{\lambda}(z)$ using closed-form formulae when the order is half of positive odd integer.

Usage

besselK_inc_clo(x, z, lambda, lower = FALSE, expon.scaled = FALSE)

Arguments

Details

One of the integral representations of $K_{\lambda}(z)$ is given by $$K_{\lambda}(z) = \sqrt{\frac{\pi}{2z}} \frac{1}{\Gamma(\lambda + \frac{1}{2})}e^{-z}\int_0^{\infty} e^{-\xi}\xi^{\lambda - 1/2}\left(1+\frac{\xi}{2 z} \right)^{\lambda -1/2}\,d\xi,$$ besselK_inc_clo evaluates closed-form formulae, which we derived to compute this integral, in the intervals $(0, x)$ and $(x, \infty)$ for the so-called lower and upper incomplete Bessel function respectively. ``Exact" evaluation of the integral in these finite intervals can also be obtained by numerical integration using software such as Maple www.maple.com.

References

Olver, F.W.J., Lozier, D.W., Boisver, R.F., Clark, C.W., 2010. Handbook of Mathematical Functions. New York: National Institute of Standards and Technology, and Cambridge University Press. Watson, G.N., 1931. A Treatise on the Theory of Bessel Functions and Their Applications to Physics. London: MacMillan and Co.

Examples

Run this code

options(digits = 15)	
## For x = 5, z = 8, lambda = 15/2 Maple 15 gives exact value of the
## lower incomplete Bessel function 0.997 761 151 460 5189(-4)
besselK_inc_clo(5, 8, 15/2, lower = TRUE, expon.scaled = FALSE)
## For x = 21, z = 8, lambda = 21/2 Maple 15 give exact value of the
## upper incomplete Bessel function 0.704 812 324 921 884 3938(-2)
besselK_inc_clo(21, 8, 21/2, lower = FALSE, expon.scaled = FALSE)