Description Usage Arguments Details Author(s) References See Also Examples
Calculates upper and lower incomplete functions of the modified Bessel function of the third kind K_{λ}(z), see details, using closed-form formulae.
1 | besselK_inc_clo(x, z, lambda, lower = FALSE, expon.scaled = FALSE)
|
x |
Limit of the integration, |
z |
Argument of the function, |
lambda |
Order, λ = (j + \frac{1}{2}) with j = 0, 1, 2, .... |
lower |
Logical. If |
expon.scaled |
Logical. If |
One of the integral representations of K_{λ}(z) is given by
K_{λ}(z) = √{\frac{π}{2z}} \frac{1}{Γ(λ + \frac{1}{2})}\,e^{-z}\int_0^{∞} e^{-ξ}ξ^{λ - 1/2}≤ft(1+\frac{ξ}{2 z} \right)^{λ -1/2}\,dξ,
besselK_inc_clo
evaluates
closed-form formulae, which we derived to compute this
integral, in the (0, x) and (x, ∞) intervals
for the so-called lower and upper incomplete Bessel
function respectively. “Exact" evaluation of the integral
in these intervals can also be obtained by
numerical integration using software such
as Maple www.maple.com.
Thanh T. Tran frmqa.package@gmail.com
Olver, F.W.J., Lozier, D.W., Boisver, R.F. and 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.
besselK_app_ser
, besselK_inc_erfc
1 2 3 4 5 6 7 | 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)
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