Can Poles of Integer Order be Eliminated in the Redefinition of Integrals?

[eljose]

Let,s suppose we have the integral:

$$\int_{-\infty}^{\infty}dxF(x)$$

but unfortunately we have a problem..the function F(x) has several poles of integer order r (r=1,2,3,4...) so it diverges :uhh: :uhh: :uhh: then my question is if there is a form to redefine our integral so it can be assigned a finite value by "eliminating" somehow its poles considering them as residues so finally we have an integral:

$$\int_{-\infty}^{\infty}dxF(x)+ Res(F)$$ where the integral is

finite and Res(F) would be the sum of the residues of F(x) at its poles...or something similar..pehaps with the "Cauchy principal value integral"?...

 

[HallsofIvy]

Of course you can (assuming a finite number of poles): place small circles around the poles and integrate over the remaining area. Of course, if your function has only poles as singularities, then its integral on this new domain is 0 so the integral on the original area is just the sum of the residues. But we knew that already, didn't we?

 

[Hurkyl]

but unfortunately we have a problem..the function F(x) has several poles of integer order r (r=1,2,3,4...) so it diverges

That's not the only reason a function may diverge. F(x) also fails to converge to zero as x goes to infinity. And even if it did go to zero, it might not do it fast enough.

Speaking about residues, AFAIK, only make sense if you're talking about complex analytic functions.

 

[eljose]

But..is there a way to "avoid" the singularities of the integrand?..