Residues of ((log(z))^2)/(1+z^2)^2

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In summary, the conversation discusses the calculation of residues for the function ((log(z))^2)/(1+z^2)^2 using methods of contour integration. The singularities at i and -i are identified and the limits are taken to calculate the residues at each point. The poles are found to be double poles and the limit formula for higher order poles is referenced on Wikipedia for further help.
  • #1
NT123
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Homework Statement

This is actually on Wikipedia, but it doesn't show how you actually calculate the residues.

I want to calculate the residues for ((log(z))^2)/(1+z^2)^2. Wikipedia claims the sum of the residues is (-(pi/4) + (i*(pi)^2)/16 - (pi/4) - (i*(pi)^2)/16) = -pi/2


Homework Equations



http://en.wikipedia.org/wiki/Methods_of_contour_integration

The Attempt at a Solution

The singularities are at i and -i. Multiplying first by (z-i)^2 and taking the limit z ---> i, I get (log(i)^2)/4 = (log(e^(i*(pi/2))^2)/4 = -(pi^2)/16.

Then, multiplying by (z+i)^2 and taking the limit z ---> -i I get (log(e^-i*(pi/2))^2)/4 = -(pi^2)/16 again. Any help would be very much appreciated.
 
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  • #2
The poles are double poles. Look at http://en.wikipedia.org/wiki/Residue_(complex_analysis ) under "Limit formula for higher order poles". There's a derivative involved here as well as a limit.
 
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1. What are residues in complex analysis?

Residues in complex analysis refer to the complex numbers that remain after a complex function is evaluated at a point where it is not defined, typically at a singularity or pole.

2. How do you calculate residues?

The formula for calculating a residue at a pole of order n is Res(f, z0) = limz→z0 [(d^(n-1)/dz^(n-1)) (z-z0)^n f(z)]. This involves taking the derivative of the function at the pole and evaluating it at that point.

3. What is the significance of residues in complex analysis?

Residues are important in complex analysis because they allow us to evaluate complex integrals using the Residue Theorem. They also help us determine the behavior of a function near a singularity or pole.

4. Can residues be negative?

Yes, residues can be negative. The sign of a residue depends on the order of the pole and the value of the function at that point.

5. How are residues used in real-world applications?

Residues have many applications in various fields such as physics, engineering, and economics. They are used to solve problems involving complex integrals, evaluate infinite series, and study the behavior of systems with poles or singularities. They are also used in the calculation of residues of residues in quantum mechanics and in the study of fluid dynamics.

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