Residue theorem problem involving real value Intergral

In summary, the integral from infinity to 0 of x^2/(x^2+1)(x^2+16) can be solved using the residue theorem as the sum of the residues times 2πi. The relevant singularities are at i and 4i, which are simple poles. By substituting these values into the simplified fractions, we can obtain the residues -2i/15 and i/30, respectively. Adding these together and multiplying by ∏i, we get the final result of ∏/10.
  • #1
xdrgnh
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Homework Statement



∫ from infinity to 0 of x^2/(x^2+1)(x^2+16)

Homework Equations



The Integral will be the sum of the residues times 2πi

The Attempt at a Solution



The only singularities that matter are at i and 4i and they are simple poles. So x^2/(x^2+1)/(2x) and I plug in 4i and get -2i/(15) then I plug in i into x^2/(x^2+16)/(2x) and get i/30 and when I add them I get -3i/30. Because it's only from 0 and to negative infinity I mult that by ∏i and I get ∏/10.
 
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  • #2
Looks good.

You glossed over several steps, which I'm assuming you understand. Just make sure you know why you can do what you did.
 

FAQ: Residue theorem problem involving real value Intergral

What is the residue theorem problem involving real value integral?

The residue theorem problem involving real value integral is a mathematical concept that involves calculating the value of an integral using the residues of a function and the Cauchy integral formula. It is a useful tool in complex analysis and has applications in various areas of science and engineering.

How does the residue theorem work?

The residue theorem states that the value of an integral over a closed contour is equal to the sum of the residues of the function within the contour. The residues can be calculated by finding the poles of the function and using the Cauchy residue formula. This allows for the evaluation of complex integrals by reducing them to simpler real integrals.

What are the key steps in solving a residue theorem problem involving real value integral?

The key steps in solving a residue theorem problem involving real value integral are:

  1. Identifying the poles of the function within the contour
  2. Determining the residues of the function at each pole
  3. Using the Cauchy residue formula to calculate the value of the integral

What are some applications of the residue theorem in science?

The residue theorem has many applications in science, including:

  • Calculating the values of complex integrals in physics and engineering
  • Studying the behavior of electric and magnetic fields in electromagnetism
  • Understanding the properties of fluids and fluid flow in fluid mechanics
  • Modeling the behavior of waves in acoustics and optics

Are there any limitations or restrictions when using the residue theorem?

Yes, there are some limitations and restrictions when using the residue theorem. These include:

  • The function must be analytic within the contour of integration
  • The contour of integration must be a closed curve
  • The poles of the function must be isolated, meaning there are no other poles or singularities within the contour

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