Solve Residue Integrals Easily with Our Residues Homework Help

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In summary, the problem involves finding the value of the integral shown, which can be rewritten as 1/2 times the integral from negative infinity to positive infinity of a function that is symmetric on the real axis. This symmetry can help in choosing a contour for integration and finding the poles. Instead of integrating over a half circle and returning along the negative real axis, one can integrate along the line z = r exp(i pi/3) and sum over the three poles in the upper half plane.
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gtfitzpatrick
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Homework Statement



show [tex]^{\infty}_{0}\int[/tex] x2 / (x6 + 1) = [tex]\pi[/tex]/6


Homework Equations





The Attempt at a Solution



= 1/2 [tex]\int^{\infty}_{-\infty}[/tex] ...

I am not sure how to go about this any pointers pls?
 
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  • #2


you're really supposed to attempt some working

but note the function is symmetric on the real axis... and the magnitude will get small as the radius gets big, this should help you pick a contour to integrate

it will help to find the poles as well...
 
  • #3


Note that you don't need to integrate over the half circle and come back along the negative real axis. You would then have to sum over the three poles in the upper half plane. Instead, you can return along the line z = r exp(i pi/3).
 

What is the purpose of solving residue integrals?

The purpose of solving residue integrals is to find the value of a complex integral that cannot be easily evaluated by traditional methods. This is done by finding the residues, or singular points, of the function and using them to calculate the integral.

Why is it important to solve residue integrals easily?

Solving residue integrals easily is important because it allows for more efficient and accurate calculations. This can save time and effort in solving complex integrals, especially in fields such as physics and engineering where these types of integrals are common.

How can I use the residues homework help to solve residue integrals?

The residues homework help provides step-by-step guidance on how to find and use residues to solve complex integrals. It also provides practice problems and solutions to help improve understanding and proficiency in solving residue integrals.

Do I need to have a strong math background to use the residues homework help?

While a strong math background can be helpful, the residues homework help is designed to be easily understandable for students with varying levels of mathematical knowledge. It provides explanations and examples to guide students through the process of solving residue integrals.

Are there any limitations to using the residues homework help for solving residue integrals?

The residues homework help is designed to assist with solving residue integrals, but it may not cover every possible scenario or variation of a problem. It is always important to consult with a teacher or tutor if you encounter difficulties in solving a particular integral.

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