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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.
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