Expanding f(z) in a Laurent Series

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Homework Help Overview

The discussion revolves around expanding the function f(z)=1/z(z-2) in a Laurent series for the annular region defined by 0<|z-3|<1. Participants are exploring the implications of the annular region on the series expansion.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply Taylor series expansions to the components of the function but questions the appropriateness of using a Laurent series for one of the terms. Other participants prompt consideration of the significance of the annular region in the context of the problem.

Discussion Status

Participants are engaged in clarifying the role of the annular region in the expansion process. Some guidance has been offered regarding the importance of understanding the region, but there is no explicit consensus on the correct approach to the series expansion.

Contextual Notes

The original poster expresses difficulty in visualizing the annular region, which may affect their understanding of the problem's requirements. The discussion highlights the need to consider the implications of the defined region on the series expansion.

Pouyan
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Homework Statement


Expand the function f(z)=1/z(z-2) in a Laurent series valid for the annual region 0<|z-3|<1

Homework Equations


I know 1/z(z+1) = 0.5(1/(z-2)) - 0.5(1/z)

Taylor for 0.5(1/(z-2)) is : ∑(((-1)k/2) * (z-3)k) (k is from 0 to ∞)For the second 0.5(1/z) the answer is a Taylor : ∑((1/6)*(-1/3)k * (z-3)k)

But why the answer for 0.5(1/z) is not 1/2(z-3) *∑((-1)k*(3/(z-3))k) (k is from 0 to ∞) ?

The correct answers for both are from Taylor but I thought for the second is a Laurent
 
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Did you consider the annular region you were given?
 
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Likes   Reactions: Pouyan
vela said:
Did you consider the annular region you were given?
Yes but I can't draw it in this page !
 
You don't need to draw it.

What's the significance of the annular region? Why did the problem bother giving it to you? If you understand that, it's the answer to your original question.
 

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