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
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e-mail: alexei.boulbitch at iee.lu

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