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Integral of unbounded function.

  • To: mathgroup at
  • Subject: Integral of unbounded function.
  • From: Wiener Zvi <win at>
  • Date: Sat, 12 Mar 1994 17:30:10 +0200

Dear MathGroupers,

I have the following problem.  If somebody has solved it 
(or a similar one) please help me to find an appropriate solution.

I need to integrate a function which is not bounded in the region of
integration, but its integral converges (it behaves like Log near +0).
So it is integrable in Lebesgue sence.

The function I am talking about is function of 2 variables x,y with
singular behavior when x==y, l - a given parameter (number).

(y-l/2)^2*BesselK[0, Abs[x-y]] 

After the integration:
Integrate[ (y-l/2)^2 (BesselK[0, Abs[x-y]], {y,0,l}, {x,0,l}]/.l->0.5
has failed.

I have tried to subtract the singular part
NLimit[ BesselK[0, Abs[z]]+  Log[ Abs[z]] ,   z->0]
exists and is equal 0.115932.

But also 

Integrate[ (y-l/2)^2 (BesselK[0, Abs[x-y]] +  Log[ Abs[x-y]]),
	{y,0,l}, {x,0,l}]/.l->0.5
was unsuccessfull:
Infinity::indet: Indeterminate expression -Infinity + Infinity encountered.

If somebody has an experience of work with 2D integration of unbounded
functions, please answer to 

Zvi Wiener.				win at
Dept. of Theor. Math.
The Weizmann Inst. of Sc.
Rehovot, 76100, Israel.

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