Find the Laurent Series for f(z)=1/(z(z-1)) Valid on 1<|z-1|<infinity

The Laurent series for f(z)=1/(z(z-1)) on 1<|z-1|<infinity is given by 1/z + 1/z^2 + 1/z^3 + ... + 1/z^n + ... + 1/(z-1) + 1/(z-1)^2 + 1/(z-1)^3 + ... + 1/(z-1)^n + ...
  • #1
g1990
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Homework Statement


Find the Laurent series for f(z)=1/(z(z-1)) valid on 1<|z-1|<infinity


Homework Equations


1/(1+a)=1-a+a^2-a^3... where |a|<1
we are not supposed to use integrals for this problem

The Attempt at a Solution


I want 1/(z-1) to be in my final answer, so I have 1/(z(z-1))=(1/(z-1))(1/(z-1))(1/(1+1/(z-1))=(*)
I can then expand the last of the three terms in (*) as 1/(1-1/(z-1))=1-(z-1)^-1+(z-1)^-2 etc.
Is this right? can I then multiply it by the first two (multiplicative) terms in (*) to get an extra (z-1)^-2 in each term of the series
 
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  • #2
Yes, that's right.
 

FAQ: Find the Laurent Series for f(z)=1/(z(z-1)) Valid on 1<|z-1|<infinity

What is the Laurent Series for f(z)=1/(z(z-1)) Valid on 1<|z-1|<infinity?

The Laurent Series for f(z)=1/(z(z-1)) valid on 1<|z-1|<infinity is given by 1/z + 1/(z-1).

What are the conditions for the Laurent Series to be valid?

The Laurent Series is valid when the function is analytic within the annulus centered at the point of expansion and the series converges absolutely within the annulus.

How do you determine the coefficients of the Laurent Series?

The coefficients of the Laurent Series can be determined by using the formula c_n = (1/2πi) ∮f(z)(z-a)^n dz, where ∮ represents a contour integral around the point of expansion a.

What is the radius of convergence for the Laurent Series of f(z)=1/(z(z-1))?

The radius of convergence for the Laurent Series of f(z)=1/(z(z-1)) is 1, which is the distance from the point of expansion (z=1) to the nearest singularity (z=0 or z=1).

What is the difference between a Laurent Series and a Taylor Series?

A Laurent Series includes both positive and negative powers of (z-a), while a Taylor Series only includes non-negative powers. Additionally, a Taylor Series is valid within a disk centered at the point of expansion, while a Laurent Series is valid within an annulus.

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