20090825, 03:12  #1 
Aug 2006
5979_{10} Posts 
Calculating a difficult sum
What are good ways to numerically calculate integrals? Background (skip, skim, or read depending on your patience):
I was trying to estimate the value of a difficult arithmetic sum recently and I'm not quite sure how to go about it.I'm just looking for general strategies, either better ways to calculate the integrals or a different approach to the sum. Because the sum includes primes it's a pain to work with numerically... but Pierre Dusart's bounds are tight enough at large numbers that I think I can make the integral approach work. But first I have to know that what I think is the integral is, in fact, the integral! Last fiddled with by CRGreathouse on 20090825 at 03:12 
20090825, 04:50  #2 
Dec 2008
7^{2}·17 Posts 
Why not evaluate the integrals formulaically using an implementation of Risch's algorithm (although Cherry's algorithmic extension to Rich's result seems more practical; but, as you know, that's not saying much ), and then evaluate the resulting formula numerically at the points you want?
If I interpret Cherry's paper correctly (which you gave me 4 months ago), he does implement his algorithm in Macsyma. Is Mathematica able to compute this integral (I know, it's a dumb question; but it's worth a shot)? Last fiddled with by flouran on 20090825 at 04:52 
20090825, 05:05  #3 
Aug 2006
3·1,993 Posts 
Neither Mathematica nor Maxima can evaluate the integral in closed form. That's not surprising: most integrals can't be, and I wouldn't have posted if it could have been.

20090825, 14:11  #4 
"William"
May 2003
New Haven
3^{2}·263 Posts 
Sometimes a change of variable in the integral helps. The integral convergence may be the same problem as the summation convergence  the integrand decreases so slowly that you need to keep extending the range of integration. The idea is to get a new integration variable that rapidly covers ranges of the old integration variable. Perhaps z = log x.
William 
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