Description Usage Arguments Details Author(s) References See Also
Evaluates analytical formulae for the lower and upper incomplete Laplace integral in terms of the complementary error function.
1 | CalIncLapInt(lambda, a = 1, b = 1, x = 1, lower = TRUE, bit = 200)
|
lambda |
Order λ = \pm(j + \frac{1}{2}) with j = 0, 1, 2, …. |
a |
Argument a > 0. |
b |
Argument b >= 0. |
x |
Limit of the incomplete Laplace integral. |
lower |
Logical. Lower incomplete Laplace integral
is returned if |
bit |
Precision bit. A positive integer greater or equal 100. |
The lower and upper extended Laplace integrals are given by
\widehat L_{λ}(x, a, b) = \int_0^{x} e^{-(a ξ^2 + b/ξ^2)}ξ^{-2λ - 1}\; dξ ,
and
\widetilde L_{λ}(x, a, b) = \int_x^{∞} e^{-(a ξ^2 + b/ξ^2)}ξ^{-2λ - 1}\; dξ
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.
Thanh T. Tran frmqa.package@gmail.com
Hankin, R.K.S (2006) Additive 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.
besselK_inc_err
, gamma_inc_err
, pgig
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