Re: Divergent integration result
- To: mathgroup at smc.vnet.net
- Subject: [mg110400] Re: Divergent integration result
- From: Alexei Boulbitch <alexei.boulbitch at iee.lu>
- Date: Wed, 16 Jun 2010 05:41:08 -0400 (EDT)
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 at 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.