Re: Reasonable integration speed? (24 hrs and counting)
- To: mathgroup at smc.vnet.net
- Subject: [mg68084] Re: Reasonable integration speed? (24 hrs and counting)
- From: axlq at spamcop.net (axlq)
- Date: Fri, 21 Jul 2006 17:36:42 -0400 (EDT)
- References: <e9nkvm$9vo$1@smc.vnet.net> <e9qabo$8bl$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <e9qabo$8bl$1 at smc.vnet.net>, Paul Abbott <paul at physics.uwa.edu.au> wrote: >1. To paraphrase Barbie (http://www.sniggle.net/barbie.php), >"integration is hard". :) That's why I downloaded a trial version of Mathematica. >2. Let's break down your problem. First, the term involving p > > Integrate[Cos[(2 n + 1) Pi x/d] p, {x, -d/2, d/2}] > >is trivial. Next, can you compute, or do you expect to be able to compute > > Integrate[Cos[(2n + 1) Pi x/d] Exp[-I k f[x]], {x, -d/2, d/2}] ? > >This integral, even though it appears to be simple, is non-trivial. I >suspect that it can be computed via Bessel functions. What I did was use Mathematica's Expand[] function to break down the integrand into individual terms. I got 13 terms, the first of which was the trival one you describe above. All the others have one of these 6 forms: c1 I Exp[-I k f] Cos[c2 x] / f^5 c1 I Exp[-I k f] x Cos[c2 x] / f^5 c1 I Exp[-I k f] x^2 Cos[c2 x] / f^5 c1 Exp[-I k f] Cos[c2 x] / f^4 c1 Exp[-I k f] x Cos[c2 x] / f^4 c1 Exp[-I k f] x^2 Cos[c2 x] / f^4 ...where c1 and c2 are constants, and f = Sqrt[a^2+(q-x)^2]. As you said, these are non-trivial to solve. Mathematica seems to get hung up on any of them. Thanks for the suggestions. I think I will have to resort to numerical integration for this one. I was hoping to get a closed-form solution. This is eventually going into some Visual Basic code, in which I have to manage the real and imaginary parts myself. -Alex