Divergent integration result
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From: Kamil on 15 Jun 2010 02:29 Hello all, I tried to evaluate the integral below, integrandnumcc = Integrate[(D[wcc, {r, 2}]^2 + (1/r D[wcc, r])^2 + 2 v 1/r D[wcc, r]*D[wcc, {r, 2}]) r, {r, 0, 1}] where wcc = (r - r^2) + n (r - r^3); What I am getting is: Integrate::idiv: Integral of -4-4 n+1/r+(2 n)/r+n^2/r+8 r-6 n r-6 n^2 r +36 n r^2+45 n^2 r^3+<<7>> does not converge on {0,10}. >> Can anybody help find what is wrong? Help will be apprecated.
From: Daniel Huber on 16 Jun 2010 05:38 Hi Kamil, your integrand has a term that is porportional to 1/r near zero and can therefore not be integrated from 0 to 10. cheers, Daniel Am 15.06.2010 08:29, schrieb Kamil: > Hello all, > > I tried to evaluate the integral below, > > integrandnumcc = > Integrate[(D[wcc, {r, 2}]^2 + (1/r D[wcc, r])^2 + > 2 v 1/r D[wcc, r]*D[wcc, {r, 2}]) r, {r, 0, 1}] > > where > wcc = (r - r^2) + n (r - r^3); > > What I am getting is: > Integrate::idiv: Integral of -4-4 n+1/r+(2 n)/r+n^2/r+8 r-6 n r-6 n^2 r > +36 n r^2+45 n^2 r^3+<<7>> does not converge on {0,10}.>> > > Can anybody help find what is wrong? > > Help will be apprecated. > >
From: Alexei Boulbitch on 16 Jun 2010 05:41
Hi, your expression staying under the integral contains the term ~1/r: wcc[r_] := (r - r^2) + n (r - r^3); dwcc[r_] := ((D[wcc[r], {r, 2}])^2 + (1/r D[wcc[r], r])^2 + (2*v)/r* D[wcc[r], r]*D[wcc[r], {r, 2}]) r // Simplify Collect[dwcc[r] // Expand, r] -4 - 4 n + (1 + 2 n + n^2)/r - 4 v - 4 n v + r^2 (36 n + 36 n v) + r (8 - 6 n - 6 n^2 + 8 v - 12 n v - 12 n^2 v) + r^3 (45 n^2 + 36 n^2 v) It is the term (1 + 2 n + n^2)/r and it will of course give rise to a logarithmic divergence on the lower integral limit (i.e. at r=0). Introducing instead the lower limit as epsilon, 0<epsilon<1 one finds Integrate[dwcc[r], {r, \[Epsilon], 1}, Assumptions -> {\[Epsilon] \[Element] Reals && \[Epsilon] > 0 && \[Epsilon] < 1}] -(1/4) (-1 + \[Epsilon]) (16 (1 + v) \[Epsilon] + 3 n^2 (1 + \[Epsilon]) (11 + 15 \[Epsilon]^2 + 4 v (1 + 3 \[Epsilon]^2)) + 4 n (5 + 9 \[Epsilon] + 12 \[Epsilon]^2 + 2 v (1 + 3 \[Epsilon] + 6 \[Epsilon]^2))) - (1 + n)^2 Log[\[Epsilon]] where the last term exhibits such a divergence. So, everything is right. Have fun, Alexei Hello all, I tried to evaluate the integral below, integrandnumcc = Integrate[(D[wcc, {r, 2}]^2 + (1/r D[wcc, r])^2 + 2 v 1/r D[wcc, r]*D[wcc, {r, 2}]) r, {r, 0, 1}] where wcc = (r - r^2) + n (r - r^3); What I am getting is: Integrate::idiv: Integral of -4-4 n+1/r+(2 n)/r+n^2/r+8 r-6 n r-6 n^2 r +36 n r^2+45 n^2 r^3+<<7>> does not converge on {0,10}. >> Can anybody help find what is wrong? Help will be apprecated. -- Alexei Boulbitch, Dr. habil. Senior Scientist Material Development IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 CONTERN Luxembourg Tel: +352 2454 2566 Fax: +352 2454 3566 Mobile: +49 (0) 151 52 40 66 44 e-mail: alexei.boulbitch(a)iee.lu www.iee.lu -- This e-mail may contain trade secrets or privileged, undisclosed or otherwise confidential information. If you are not the intended recipient and have received this e-mail in error, you are hereby notified that any review, copying or distribution of it is strictly prohibited. Please inform us immediately and destroy the original transmittal from your system. Thank you for your co-operation. |