CalIncLapInt: Evaluation of Analytical Formulae for the
Incomplete Laplace Integral
Description
Evaluates analytical formulae for the
lower and upper incomplete Laplace integral in terms of the
complementary error function.Usage
CalIncLapInt(lambda, a = 1, b = 1, x = 1, lower = TRUE, bit = 200)
Details
The lower and upper extended Laplace integrals
are given by $$\widehat L_{\lambda}(x, a, b) = \int_0^{x}
e^{-(a \xi^2 + b/\xi^2)}\xi^{-2\lambda - 1}\; d\xi ,$$ and
$$\widetilde L_{\lambda}(x, a, b) = \int_x^{\infty}
e^{-(a \xi^2 + b/\xi^2)}\xi^{-2\lambda - 1}\; d\xi$$ respectively.
Calculation is performed using multiple precision
floating-point reliably or MPFR-numbers instead of the
default floating-point number in R, which ensure
accuracy is at least 100 bit.References
Hankin, R.K.S (2006) {A}dditive integer partitions in {R}.
Journal of Statistical Software, Code Snippets, 16.
Maechler, M Rmpfr: R MPFR - Multiple Precision Floating-Point
Reliable. R package version 0.5-0, http://CRAN.R-project.
org/package=Rmpfr.
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.
Tran, T. T., Yee, W.T. and Tee, J.G (2012) Formulae for
the Extended Laplace Integral and Their Statistical
Applications. Working Paper.
Watson, G.N (1931) A Treatise on the Theory of
Bessel Functions and Their Applications to
Physics. London: MacMillan and Co.See Also
besselK_inc_err, gamma_inc_err, pgig