Multi-dimensional residue theorem

1. Feb 25, 2008

Pere Callahan

Hi,

I'm wondering if a generalization of the residue theorem/formulae to several complex variables could be just as helpful as in the one-dimensional case.

For example if you were to calculate

$$\int_\mathhb{R}{\frac{dk}{2\pi}\frac{e^{-ikx}}{1+k^2}}$$

One way would be to observe that the integrand has simple poles at $$k=\pm i$$ then close the contour of integration in the upper (lower) halfplane when x is less (greater) then zero, to obtain the result

$$e^{-|x|}$$.

What however in the slightly more complicated case

$$\int_{\mathhb{R}^2}{\frac{d^2k}{(2\pi)^2}\frac{e^{-ik_1x}e^{-ik_2y}}{1+k_1^2+k_2^2}}$$

The integrand has "poles" (are they still called that if they are not isolated?) for $$k_1^2+k_2^2 = -1$$ which might be thought of as a circle in the "imaginary plane".

Is there a similar way to evaulate this integral by somehow closing the contour and applying some generalization of the residue theorem?

If not, how COULD it be calculated? My thought was to do the, say, k_1 integration first. For fixed k_2, there are poles at

$$k_1^\pm = \pm i \sqrt{1+k_2^2}$$

with residues

$$Res^\pm = \frac{e^{\pm\sqrt{1+k_2^2} x}e^{-ik_2y}}{\pm 2 i \sqrt{1+k_2^2}}$$
where we have to pick the right sign according to whether x is positive or negative.

Then I would do the k_2 integration. The poles are at

$$k_2^\pm = \pm i$$

However

$$\frac{e^{\pm\sqrt{1+k_2^2} x}e^{-ik_2y}}{\pm 2 i \sqrt{1+k_2^2}}$$

is not holomorphic in a neighbourhood of $$\pm i$$ so I cannot apply the Residue theorem can I?

Thanks

-Pere

Last edited: Feb 26, 2008
2. Feb 26, 2008

mathwonk

for a nice treatment of residue in higher dimensions see griffiths paper, on certain rational integrals, in annals of math, about 30 years ago or more. or look in griffiths harris book principles of algebraic geometry.

the point is to replace the integral of a meromorphic form with a pole along a divisor, by the integral of a different form, the residue form, over the codimension one set supporting the pole.

3. Feb 26, 2008

Pere Callahan

OK I have the paper and will read it tonight, although my current understanding of Algebraic Geometry is fairly limited ...

Is there, for the special integral I posted above, a direct way of evaluating it...?

-Pere

Last edited: Feb 26, 2008
4. Mar 8, 2008

Pere Callahan

Well. it seems to be too limited to understand what Griffiths is talking about in that paper (as well as in the book) Can the method be applied to the special integral posted above without knowing too much about the theory behind? Maybe somebody could lead me through this example..? (This is asked a lot, I know)

Thanks

-Pere