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logarithmic integrals in R?

4 messages · Hans W Borchers, Oliver Kullmann

Original
Oliver Kullmann
# 
Hello,

I couldn't find information on whether the logarithmic integrals

Li_m(x) = integral_0^x log(t)^(-m) dt

for x >= 0 are available in R?

Best wishes

Oliver
9 days later
Hans W Borchers
# 
Oliver Kullmann <O.Kullmann <at> swansea.ac.uk> writes:
I saw your request only this weekend.
The first logarithmic integral can be computed using the exponential
integral Ei(x) per

    li(x) = Ei(log(x))

and elliptic integrals are part of the 'gsl' package, so

    library('gsl')
    x <- seq(2, 10, by=0.5)
    y <- expint_Ei(log(x))
    y

See e.g. the Handbook of Mathematical Functions for how to reduce higher
logarithmic integrals.
Another possibility is to use the Web API of 'keisan', the calculation
library of Casio.

Regards
Hans Werner
Oliver Kullmann
# 
Thanks for the information.
On Sat, May 29, 2010 at 01:15:29PM +0000, Hans W. Borchers wrote:
I found gsl at http://cran.r-project.org/web/packages/gsl/index.html.
However here I wasn't succesful: Going through the chapter

http://www.math.ucla.edu/~cbm/aands/page_228.htm

I didn't find any mentioning of the higher logarithmic integrals.
Interesting; but again only li(x).

Also a google search on "higher logarithmic integrals", "logarithmic integrals"
or "li_n(x)" doesn't reveal anything, so I would be thankful for a hint.

Thanks again!

Oliver
Hans W Borchers
# 
Oliver Kullmann <O.Kullmann <at> swansea.ac.uk> writes:
What I meant were formulas like

    \int 1/\log(t)^2 dt = -t/\log(t) + li(t)

    \int 1/\log(t)^3 dt = 1/2 * ( -t/\log(t)^2 - t/\log(t) + li(t) )

and higher forms that can be expressed through the Gamma function.
I am certain I 've seen them in AandS' handbook (where else?), but
sure cannot remember in which chapter or page.

Which logarithmic integrals do you really need, and on what range?

Regards,
Hans Werner
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