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Partial Fractions in Laurent Series Expansion

  • Thread starter tadf2
  • Start date
  • #1
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Homework Statement


f = [itex]\frac{1}{z(z-1)(z-2)}[/itex]


Homework Equations


Partial fraction



The Attempt at a Solution


R1 = 0 < z < 1
R2 = 1 < z < 2
R3 = z > 2


f = [itex]\frac{1}{z(z-1)(z-2)}[/itex] = [itex]\frac{1}{z}[/itex] * ([itex]\frac{A}{z-1}[/itex] + [itex]\frac{B}{z-2}[/itex])
Where A = -1 , B = 1.
f = [itex]\frac{1}{z}[/itex] * ([itex]\frac{1}{z-2}[/itex] + [itex]\frac{1}{1-z}[/itex])

My question is, why do you have to use partial fractions?

Can't you just leave the initial multiplication of three terms and expand them individually;
[itex]\frac{1}{z-1}[/itex] (dividing 1/z to numerator and denominator at R2,R3)
and
[itex]\frac{1}{z-2}[/itex] (dividing 1/z to numerator and denominator at R3)

so for example,

for R2,
f = [itex]\frac{1}{z}[/itex] * [itex]\frac{1}{z-2}[/itex] * (-[itex]\frac{1}{z}[/itex]*[itex]\frac{1}{1-z^{-1}}[/itex])

You can expand 1/(1-z) and 1/(z-2) using taylor series.

Should you always use partial fractions?
Or does expanding without using partial fractions still work?
]

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
STEMucator
Homework Helper
2,075
140
I believe your partial fraction expansion is incorrect, you should check it again.

##\frac{A}{z} + \frac{B}{z-1} + \frac{C}{z-2}##
 
  • #3
5
0
[itex]\frac{A}{z}[/itex] this part is not necessarily needed for partial fractions because
it doesn't require any expansions..
 
  • #4
STEMucator
Homework Helper
2,075
140
[itex]\frac{A}{z}[/itex] this part is not necessarily needed for partial fractions because
it doesn't require any expansions..
You don't really even need partial fractions at all. You could write it like this:

##(\frac{1}{z})(\frac{1}{z-1})(\frac{1}{z-2})##

Then write the terms as their respective series, for example:

##\frac{1}{z} = \sum_{n=0}^{∞} (-1)^n (-1+z)^n ## where ##|1-z|<1##

I believe that's what you were asking anyway.
 
  • #5
5
0
Cool, I get it now. You answered my question.

Thanks Zondrina!
 

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