Partial Fractions in Laurent Series Expansion

In summary, the conversation discussed the use of partial fractions in simplifying a complex equation. The equation was expanded using partial fractions, but it was suggested that it may not be necessary and the equation could be written in a simpler form using series.
  • #1
tadf2
5
0

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?
]
 
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  • #2
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
[itex]\frac{A}{z}[/itex] this part is not necessarily needed for partial fractions because
it doesn't require any expansions..
 
  • #4
tadf2 said:
[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
Cool, I get it now. You answered my question.

Thanks Zondrina!
 

What is a partial fraction in Laurent series expansion?

A partial fraction in Laurent series expansion is a mathematical technique used to express a rational function as a sum of simpler fractions. It is commonly used in complex analysis to simplify complex functions and allow for easier integration and differentiation.

How is a partial fraction in Laurent series expansion calculated?

To calculate a partial fraction in Laurent series expansion, the function must first be expressed as a polynomial divided by a polynomial. Then, using algebraic manipulation, the denominator is factored into linear and quadratic terms. Finally, the coefficients of the partial fractions are solved for using a system of equations.

Why is partial fraction in Laurent series expansion useful?

Partial fraction in Laurent series expansion is useful because it allows for the integration and differentiation of complex functions. It also helps to simplify complex functions, making them easier to work with and understand.

What are some applications of partial fraction in Laurent series expansion?

Partial fraction in Laurent series expansion has many applications in mathematics, physics, and engineering. It is commonly used in solving differential equations, evaluating integrals, and in signal processing. It is also used in the study of complex functions and in the analysis of electrical circuits.

Are there any limitations to using partial fraction in Laurent series expansion?

One limitation of partial fraction in Laurent series expansion is that it can only be applied to rational functions. It also requires knowledge of algebraic manipulation and solving systems of equations, which can be challenging for some. Additionally, the method may not always yield a simple or concise result.

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