Evaluating Integrals using the Residue THM

In summary, the function sinz/z2 has a simple pole at z = 0. The rules state that if a function has a simple pole at z0, the residue is equal to the limit as z approaches z0 of (z - z0) * f(z). Integrating first is not necessary. The residue can also be calculated using the other rules, but they will give the same result in this case.
  • #1
brianhawaiian
12
0

Homework Statement



integral |z|=1 of sinz/z2dz


Homework Equations


Rule #1 if f(z) has a simple pole at z0, then
Res[f(z),z0] = lim(as z goes to z0) (z - z0)*f(z)

Rule #2 if f(z) has a double pole at z0, then
Res[f(z),z0] = lim(as z goes to z0)d/dz (z - z0)2*f(z)

Rule #3 If f(z) and g(z) are analytic at z0, and if g(z) has a simple zero at z0 then,
Res[f(z)/g(z), z0] = f(z0)/g'(z0)

Rule #4 If g(z) is analytic and has a simple zero at z0 then,
Res[1/g(z), z0] = 1/g'(z0)





The Attempt at a Solution



Just confused on which one to use? Would I integrate first? And if so what would my z0 be in Rest[..., z0]

Thanks
 
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  • #2
What kind of of pole does the function sinz/z2 have and where is it? Although I think 1 and 2 will give the same result.
 
  • #3
phsopher said:
What kind of of pole does the function sinz/z2 have and where is it? Although I think 1 and 2 will give the same result.

double pole at i/2 and -i/2?

Edit: Actually simple pole at z = 0?
 
Last edited:
  • #4
Yep.
 

1. What is the Residue Theorem and how is it used to evaluate integrals?

The Residue Theorem is a mathematical concept that allows for the evaluation of certain types of integrals. It states that if a function is analytic in a closed region except for finitely many isolated singularities, the integral of that function along a closed contour is equal to the sum of the residues of the function at its singularities within the contour. This allows for the evaluation of integrals that may be difficult or impossible to solve through other methods.

2. What are singularities and how do they relate to the Residue Theorem?

Singularities are points in a function where it is either undefined or has a discontinuity. In the context of the Residue Theorem, singularities are important because they are the only points where the integral will have a non-zero value. The Residue Theorem states that the integral is equal to the sum of the residues at these singularities, making them crucial in evaluating the integral.

3. What is a residue and how is it calculated?

A residue is the value of a function at a singularity. In order to calculate a residue, one can use the formula Res(f,c) = lim(z→c) [(z-c)f(z)], where c is the singularity and f(z) is the function. In other words, the residue is found by taking the limit of (z-c)f(z) as z approaches the singularity.

4. What types of integrals can be evaluated using the Residue Theorem?

The Residue Theorem is primarily used for integrals around closed contours of functions with isolated singularities. This includes integrals of rational functions, trigonometric functions, exponential functions, and many others. However, not all integrals can be evaluated using the Residue Theorem, as it relies on the function being analytic (i.e. differentiable) within the contour.

5. Are there any limitations or drawbacks to using the Residue Theorem?

While the Residue Theorem is a powerful tool for evaluating certain types of integrals, it does have its limitations. One limitation is that it only applies to functions with isolated singularities, so it cannot be used for integrals with continuous singularities. Additionally, the process of finding residues can be time-consuming and may require advanced mathematical techniques. Lastly, the Residue Theorem may not always provide an exact solution and may only yield an approximation.

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