Problem with a contour integral

In summary, The conversation is about solving a complex integral with the equation Integral from 0 to infinity of{dx((x^2)(Sin[xr])}/[((x^2)+(m^2))x*r], where m and r are real variables. The person tried to solve it by choosing a half "donut" in the upper part of the plane with radii or p and R, and then breaking up the contour integral into paths 1, 2, 3, and 4. They ask if this is the correct approach and later state that they believe they have found the correct solution by finding the residue at (im). They also confirm that (-im) is not in the contour and R(0)=
  • #1
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I was w=kind of confused as of how to go about solving this integeal using complex methods. it is the Integral from 0 to infinity of{dx((x^2)(Sin[xr])}/[((x^2)+(m^2))x*r] where m and r are real variables. I tried to choose a half "donut" in the upper part of the plane with radii or p and R. Then I tried to break up the contour integral into paths 1, 2, 3, 4. is this the correct way to go about this? Thanks
 
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  • #2
I think that i figured it out... I found the residue at (im) as this is the only residue that contributes to the integral (-im is not in the contour and R(0)=0). Does this sound right? Thanks
 

1. What is a contour integral?

A contour integral is a type of integral used in complex analysis to calculate the value of a function along a specific path in the complex plane. It is also known as a line integral or path integral.

2. What is the problem with a contour integral?

The main problem with a contour integral is that it is highly dependent on the chosen path or contour. If the chosen path is not well-defined or contains singularities, the contour integral may not accurately represent the value of the function.

3. How do you choose the appropriate contour for a given integral?

Choosing the appropriate contour for a given integral requires a thorough understanding of the function, its singularities, and the desired outcome. In general, the contour should avoid any singularities and enclose the region of interest.

4. What are some common techniques for evaluating contour integrals?

Some common techniques for evaluating contour integrals include the method of residues, Cauchy's integral formula, and Cauchy's integral theorem. These methods involve using complex analysis and calculus to simplify the integral and find its value.

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

Yes, contour integrals have many applications in physics, engineering, and other areas of science. They can be used to solve problems involving electric and magnetic fields, fluid flow, and quantum mechanics, among others.

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