Help Needed - Region of Convergence for Laurent Series of f(z)

In summary, the conversation discusses finding the region of convergence for a Laurent series. The function in question is f(z) = 1/[(z^3)*cosz] and the Laurent series converges at Pi/4. The participants mention the need for the function to be analytic for the series to converge, and the exceptions to this rule. Ultimately, it is determined that the Laurent series converges for 0< |z|< \pi/2.
  • #1
Butelle
12
0
Hello -

I have a problem in general finding the region in which the Laurent series converges...

Could someone please help me with this question - I know that this is is meant to be easy (as there is no fully worked solution to this) but I don't understand it:

f(z) = 1/ [(z^3)*cosz]. The function has a Laurent series about z = 0 converging at Pi/4. What is the region in which this Laurent series converges?

Thanks.
 
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  • #2
When your textbook derived the Laurent expansion, what was the shape of the domain that the function had to be analytic in (it is complex analysis after all - isn't everything about being analytic?). What can cause a function to not have that property, and hence will make the series not converge?
 
  • #3
I'm sorry I am not really understanding? I derived the expansion and then I didn't know what to do after that?
 
  • #4
He is saying that the series will converge until it hits a point at which the function is NOT analytic. This particular function, f(z)= 1/(z3cos(z)), is analytic except where the denominator is 0: z= 0 or z an odd multiple of [itex]\pi/2[/itex]. That is, the Laurent series around z= 0 converges for 0< |z|< [itex]\pi/2[/itex].
 
  • #5
ahhh ok i understand - see i didnt even understand the theory behind it! i get it now - thanks! :)
 

1. What is the Region of Convergence (ROC) for a Laurent Series?

The ROC for a Laurent Series is the set of all complex numbers for which the series converges. In other words, it is the set of all points in the complex plane where the series of the function is well-defined and has a finite value.

2. How is the ROC determined for a Laurent Series?

The ROC for a Laurent Series is determined by finding the largest annulus in the complex plane where the function is analytic and has no singularities. This annulus is called the annulus of convergence and its boundaries determine the ROC.

3. Can the ROC for a Laurent Series be an empty set?

Yes, it is possible for the ROC of a Laurent Series to be an empty set. This occurs when the function has singularities everywhere in the complex plane, meaning there is no region where the series converges.

4. What happens if a point lies on the boundary of the ROC?

If a point lies on the boundary of the ROC, the convergence of the Laurent Series at that point is unclear. It is possible that the series may converge or diverge at that point, depending on the specific function and the points chosen on the boundary.

5. How does the ROC of a Laurent Series affect its convergence in the complex plane?

The ROC of a Laurent Series directly affects its convergence in the complex plane. If a point lies within the ROC, then the series will converge at that point. If a point lies outside the ROC, then the series will diverge at that point. Furthermore, the ROC also determines the maximum radius of convergence for the series.

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