Multi-dimensional residue theorem

In summary: Algebraic Geometry is fairly limited ...A generalization of the residue theorem/formulae could be helpful in the one-dimensional case, but it is not helpful in the slightly more complicated case. There are "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". Unfortunately, this method is not helpful in the case of the special integral posted above because the integrand is not holomorphic in a neighbourhood of \pm i.
  • #1
Pere Callahan
586
1
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

[tex]\int_\mathhb{R}{\frac{dk}{2\pi}\frac{e^{-ikx}}{1+k^2}}[/tex]

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

[tex]e^{-|x|}[/tex].

What however in the slightly more complicated case

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

The integrand has "poles" (are they still called that if they are not isolated?) for [tex]k_1^2+k_2^2 = -1[/tex] 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

[tex]k_1^\pm = \pm i \sqrt{1+k_2^2}[/tex]

with residues

[tex]Res^\pm = \frac{e^{\pm\sqrt{1+k_2^2} x}e^{-ik_2y}}{\pm 2 i \sqrt{1+k_2^2}}[/tex]
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

[tex]k_2^\pm = \pm i[/tex]

However

[tex]\frac{e^{\pm\sqrt{1+k_2^2} x}e^{-ik_2y}}{\pm 2 i \sqrt{1+k_2^2}}[/tex]

is not holomorphic in a neighbourhood of [tex]\pm i[/tex] so I cannot apply the Residue theorem can I?

Thanks

-Pere
 
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  • #2
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
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
 
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  • #4
Pere Callahan said:
although my current understanding of Algebraic Geometry is fairly limited ...


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:smile:)

Thanks

-Pere
 

Related to Multi-dimensional residue theorem

1. What is the Multi-dimensional residue theorem?

The Multi-dimensional residue theorem is a mathematical theorem used in complex analysis to evaluate integrals over closed contours in multiple complex variables. It is an extension of the Cauchy residue theorem, which applies to integrals in a single complex variable.

2. How does the Multi-dimensional residue theorem differ from the Cauchy residue theorem?

The Multi-dimensional residue theorem takes into account the singularities of a function in multiple complex variables, while the Cauchy residue theorem only considers singularities in a single complex variable. This allows for the evaluation of integrals over more complex contours in higher dimensions.

3. What is the significance of the Multi-dimensional residue theorem?

The Multi-dimensional residue theorem is an important tool in complex analysis as it allows for the evaluation of integrals that may be difficult or impossible to solve using other methods. It also has applications in other branches of mathematics, such as algebraic geometry and number theory.

4. How is the Multi-dimensional residue theorem used in practical applications?

The Multi-dimensional residue theorem is used in various fields of science and engineering, such as physics, chemistry, and signal processing, to solve integrals that arise in these areas. It is also used in computer algorithms for efficient calculation of integrals in multiple complex variables.

5. Are there any limitations to the Multi-dimensional residue theorem?

Like any mathematical theorem, the Multi-dimensional residue theorem has its limitations. It is only applicable to functions that are analytic, meaning they can be represented by a convergent power series. It also assumes that the contour of integration is closed and does not intersect any singularities of the function.

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