Finding Residue using Cauchy's Integral & Residue Thms

In summary, the conversation discusses using the residue theorem and Cauchy's integral formula to calculate the residue of a function at z=0, with the conclusion that the residue can be found by taking the limit of (z-c)*f(z) for simple poles at z=c.
  • #1
brandones
3
0

Homework Statement



Calculate

[tex]\stackrel{Res}{z=0}(\frac{z^5}{(z^2-4)(z-4))}[/tex]

Homework Equations



I've learned enough Latex for one day, thank you very much. Wikipedia:

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

http://en.wikipedia.org/wiki/Cauchy's_integral_formula

The Attempt at a Solution



[tex]\stackrel{Res}{z=0}(\frac{z^5}{(z^2-4)(z-4))}[/tex]

via Cauchy's Residue Theorem:
= [tex]\frac{1}{2πi}\int \frac{z^5}{(z-2)(z+2)(z-4)} dz [/tex]

which, via Cauchy's Integral Formula:
= [tex](\frac{z^5}{(z+2)(z-4)})[/tex] solved at z=2

= [tex]\frac{32}{-8}[/tex]

=-4

Is this legit? This seems way too convenient, and weird that there isn't an explicit theorem saying I can do this.
 
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  • #2
Oh wait, I just went a really roundabout way of defining the residue, didn't I?

Well, now it seems totally obvious.

Hooray!
 
  • #3
brandones said:

Homework Statement



Calculate

[tex]\stackrel{Res}{z=0}(\frac{z^5}{(z^2-4)(z-4))}[/tex]

You mean at z=2, right?

Oh wait, I just went a really roundabout way of defining the residue, didn't I?
Yeah, for simple poles at z=c, the residue is just the limit of (z-c)*f(z).
 

1. What is Cauchy's Integral & Residue Theorems?

Cauchy's Integral Theorem is a fundamental theorem in complex analysis that states that if a function is analytic inside and on a simple closed contour, then the integral of the function around the contour is equal to the sum of the residues of the function at all of its isolated singularities inside the contour. The Residue Theorem is a powerful consequence of Cauchy's Integral Theorem, which states that the integral of a function around a closed contour is equal to 2πi times the sum of the residues of the function at all of its poles inside the contour.

2. How do I find the residues of a function?

To find the residues of a function, you must first find the singularities of the function, which are the points where the function is not analytic. Then, use the formula Res(a) = limz→a(z-a)f(z), where a is the singularity and f(z) is the given function. Alternatively, you can use the Cauchy's Integral Theorem to find the residues by finding the coefficient of the (z-a)-1 term in the Laurent series expansion of the function.

3. Can Cauchy's Integral & Residue Theorems be used for any function?

No, Cauchy's Integral & Residue Theorems can only be used for analytic functions, which means they must be differentiable at every point in their domain. Functions that have singularities or poles cannot be used with these theorems.

4. What is the purpose of finding residues using Cauchy's Integral & Residue Theorems?

Finding residues using Cauchy's Integral & Residue Theorems allows us to evaluate complex integrals that may be difficult or impossible to compute using traditional methods. It also helps us to understand the behavior of functions near their singularities and poles.

5. Are there any real-world applications of Cauchy's Integral & Residue Theorems?

Yes, Cauchy's Integral & Residue Theorems have many real-world applications in physics, engineering, and other fields. For example, they can be used to solve problems in fluid dynamics, electromagnetism, and signal processing. They are also used in the development of algorithms for numerical analysis and computational mathematics.

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