Integrating a Circle: Contour Integration Technique

In summary, the conversation discusses the possibility of integrating a circle using contour integration technique. The concept of Green's theorem is mentioned as a way to connect line/contour integration in 2D with surface integration. The main question is whether it is possible to integrate a circle using this technique and how to do it. The conversation also touches upon the difference between integrating a function of two variables over the surface of a circle and integrating around the circumference of a circle.
  • #1
julian92
24
0

Homework Statement



integrating a circle ,,
my main question is that, can we integrate it by contour integration technique ?
and if yes ,, would you please show me how :) or just give me a hint :D

Thanks is advance :-)

Homework Equations



y^2 + x^2 = a^2

where a= r

suppose that a = 2

The Attempt at a Solution



i know that it can be done as two semicircles
taking a substitution x=2sin(u) ,, or even by integration by parts!
 
Last edited:
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  • #2
integrating a circle = integrating to find the surface of a circle?
You should check out Green's theorem, it connects line/contour integration in 2D with surface integration.
 
  • #3
justsof said:
integrating a circle = integrating to find the surface of a circle?
You should check out Green's theorem, it connects line/contour integration in 2D with surface integration.

thanx for the reply :smile:
well ,, the thing is that I'm not really good at contour integration ,, I've been searching for a text to study contour integration for ages ,, and still can't find one with good details and examples

and still don't know when to use contour integration!

and since circles can't be integrated using the normal integration techniques ,, i wondered if it could be done using contour!

and i got stuck at this problem ,, and really want to integrate that little circle :(
 
  • #4
What do you mean by "integrating a circle"? Integrating a function of two variables over the surface of a circle? Integrating around the circumference of a circle?
 
  • #5
HallsofIvy said:
What do you mean by "integrating a circle"? Integrating a function of two variables over the surface of a circle? Integrating around the circumference of a circle?

I'm really not sure about the difference of the two :S

Does each one have a different method of integration?

Thanks in advance :)
 

What is "Integrating a Circle: Contour Integration Technique"?

"Integrating a Circle" or "Contour Integration Technique" is a mathematical method used to evaluate integrals, specifically those that involve complex numbers. It involves integrating along a closed curve, usually a circle, in the complex plane.

Why is the contour chosen as a circle?

The contour is chosen as a circle because it simplifies the integration process and allows for the use of the Cauchy's integral formula, which is essential in the contour integration technique. Additionally, integrating along a circle results in a closed contour, making it easier to evaluate the integral using the Residue Theorem.

What are the steps involved in integrating a circle?

The steps involved in integrating a circle using the contour integration technique are as follows:

  1. Choose a suitable contour, usually a circle, in the complex plane.
  2. Parametrize the contour using a complex variable, such as z = re^(it), where r is the radius and t is the angle parameter.
  3. Substitute the parametrized contour into the given integral, resulting in a complex-valued function of t.
  4. Evaluate the integral using the Cauchy's integral formula or the Residue Theorem.
  5. Simplify the result and take the limit as the radius of the circle approaches infinity, if necessary.

What are the applications of contour integration in science?

Contour integration is widely used in physics, engineering, and other scientific fields to solve complex integrals. It is particularly useful in solving problems involving electric and magnetic fields, fluid dynamics, and quantum mechanics. It is also used in signal processing and image reconstruction.

What are some common challenges in integrating a circle using the contour integration technique?

Some common challenges in integrating a circle using the contour integration technique include choosing the appropriate contour, parametrizing the contour correctly, identifying and evaluating the singularities, and simplifying the resulting integral. It may also be challenging to find the correct limits of integration and to take the limit as the radius of the circle approaches infinity. It is important to have a strong understanding of complex analysis and the properties of complex functions to successfully integrate a circle using the contour integration technique.

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