How can the method of residues be used to solve integrals?

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In summary, the conversation discusses finding the integral of 1/(x^4+x^2+1) from 0 to infinity using the method of residues. The individual initially struggled with finding the poles and solving the quartic equation, but eventually realized that using the substitution y=x^2 and the quadratic formula would make it easier. The final solution is (pi*sqrt(3))/6.
  • #1
jnazor
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[SOLVED] Integral using residues

Homework Statement


Show integral of 1/(x^4+x^2+1) from 0 to infinity equals (pi*sqrt(3))/6 using method of residues



Homework Equations





The Attempt at a Solution


Tried to solve x^4+x^2+1=0. Don't get very useful numbers for poles.
 
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  • #2
Well what do you know about the method of residues? You've not really shown any work now, have you?
 
  • #3
don't worry i worked it out. I didn't solve the quartic properly to get the poles before. I got my 4 poles two of which I am parts are more than 0. I evaluated the residues at those two poles and got the value. Since it's even I can divide that number by two to get the integral from 0 to inf rather than -inf to inf. I didn't notice there was a much easier way to solve that quartic by letting y=x^2 and then using quadratic formula to find y then x would be just the sqrt of y.
 
  • #4
Well done!
 

FAQ: How can the method of residues be used to solve integrals?

What is the concept of "residues" in integral calculus?

Residues refer to the complex numbers that are left over after a function is integrated along a closed contour. They play a crucial role in evaluating integrals using the residue theorem, which states that the integral of a function around a closed contour is equal to the sum of its residues inside the contour.

How do you find the residues of a function?

To find the residues of a function, you first need to identify the singular points (poles) of the function within the contour of integration. Then, you can use the residue formula to calculate the residue at each pole. This formula involves taking the limit of the function as it approaches the pole and multiplying it by a factor dependent on the order of the pole.

What is the residue theorem and how is it used?

The residue theorem is a powerful tool in complex analysis that states that the integral of a function around a closed contour is equal to the sum of its residues inside the contour. This theorem is used to evaluate complex integrals that would otherwise be difficult or impossible to solve using traditional methods.

Can the residue theorem be applied to any contour?

Yes, the residue theorem can be applied to any contour as long as the function is analytic (i.e. has no singularities) inside the contour. However, it is most commonly used for contours that form simple closed loops, such as circles or rectangles.

What are some real-world applications of using residues in integrals?

Residues are commonly used in fields such as physics and engineering to solve problems involving complex variables. They can also be applied in signal processing, fluid dynamics, and quantum mechanics, among others. In particular, they are useful for calculating the values of certain integrals that arise in these fields, making them an important tool for solving complex problems.

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