Calculation of a certain type of contour integral

In summary, the conversation discusses the computation of a contour integral with a pole on the boundary. It is suggested to have the contour run around the pole in a small semi-circular arc to get the half-residue contribution. The same idea is used in deriving the solution of the Benjamin-Ono equation. However, if the pole is of higher order, the calculation may not work. A reference is provided for further information on this topic.
  • #1
hunt_mat
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Hi,

During my research I came across a contour integral where the pole was on the boundary. I have never come across this before, do anyone of you know how I would go about computing this?

It involved the Hilbert transform and I can't find it in my undergraduate complex analysis books and I thought someone here might know.
On a similar note, if anyone has any good numerical routines for Hilbert transform, I would like to know.

Mat
 
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  • #2
The way such a thing is dealt with is to have the contour run around the pole in a small semi-circular arc, in the limit that the radius of the arc tends to zero.

As long as it's a simple pole, the result is if a contour runs straight through a pole (which would correspond to a half-circle arc that excludes the pole from being inside the contour) the pole contributes half the residue to the integral. If the pole is at a corner of a contour (which corresponds to a quarter-circle arc around the contour) it would contribute 1/4 the residue.
 
  • #3
Do you have a reference? The same idea is used in deriving the solution of the Benjamin-Ono equation.

Surely if the pole was outside of the contour the integral would be zero by Cauchy's theorem. Am I missing something?
 
  • #6
hunt_mat said:
Do you have a reference? The same idea is used in deriving the solution of the Benjamin-Ono equation.

Surely if the pole was outside of the contour the integral would be zero by Cauchy's theorem. Am I missing something?

No, it's not zero because in the limit that the arc around the pole goes to zero the contour goes straight through the pole, so there is a contribution.

Consider:

[tex]\oint_c dz \frac{f(z)}{z-z_0}[/tex]
where f(z) is analytic everywhere inside and on the contour. If the pole of the integrand at z_0 lies on our contour C, we go around it in a semi-circular arc of radius [itex]\epsilon[/itex], as [itex]\epsilon \rightarrow 0[/itex]. This part of the curve we'll parametrize by [itex]z-z_0 = \epsilon \exp(i\theta)[/itex]. This gives

[tex]\int_\pi^0 d\theta i\epsilon e^{i\theta} \frac{f(z_0 + \epsilon e^{i\theta})}{\epsilon e^{i\theta}}[/tex]

If we now take the limit [itex]\epsilon \rightarrow 0[/itex], we get
[tex]-i\int_0^\pi d\theta f(z_0) = -i\pi f(z_0)[/tex]
which is half of the residue of f(z) at z_0 (with a minus sign because we traversed over the pole in a clockwise rotation).

Note that this calculation wouldn't work if the pole weren't simple, i.e., if the pole were [itex](z-z_0)^m[/itex], with [itex]m > 1[/itex].

(However, jackmell's link suggests that if the pole is an odd power and our contour through it is a straight line segment, then treating the straight line segment as a principal value integral gives the half-residue again.)

jackmell said:
Hello Mute. Perhaps you would find this thread interesting which suggests we can do similar calculations with higher-ordered poles. See post #6 by JMerry.

http://www.artofproblemsolving.com/...67&t=182057&hilit=mainstream+complex+analysis

That is interesting. I had not heard of this result before. Also interesting is that the contour through the pole in this case must be a straight line segment.
 
Last edited:

1. What is a contour integral?

A contour integral is a type of integral used in complex analysis that integrates a function along a certain curve or path in the complex plane. It is also known as a line integral or path integral.

2. How is a contour integral calculated?

A contour integral is calculated by breaking the curve or path into small segments, approximating each segment with a linear function, and then summing the contributions from each segment. This process is known as the Riemann sum and is used to approximate the value of the integral.

3. What is the significance of contour integrals in scientific research?

Contour integrals are used in many areas of science, including physics, engineering, and mathematics. They are particularly useful in complex analysis, where they can be used to solve problems involving complex functions and complex numbers.

4. How do contour integrals differ from regular integrals?

Contour integrals are calculated in the complex plane, while regular integrals are calculated in the real plane. This means that contour integrals can involve complex functions and complex numbers, while regular integrals only involve real functions and real numbers.

5. Can contour integrals be used to solve real-world problems?

Yes, contour integrals are used in many real-world applications, such as calculating electric fields in physics, analyzing fluid flow in engineering, and evaluating probability distributions in mathematics. They are a powerful tool for solving problems that involve complex functions and complex numbers.

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