# Partial Fractions: Expanding 1/(1+z^3)^2

• jameson2
In summary, the conversation discusses the use of partial fractions to solve a question involving the expansion of f(z)=\frac{1}{(1+z^3)^2}. The fact that the bottom is squared is causing confusion and the person is unsure if they should use the complex roots. The solution is to use the identity 1+z^3=(1+z)(1-z+z^2) and then do the partial fraction as usual. The conversation also mentions a previous question about the "principal part" or "Laurent series" for 1/(z^2+1)^2, which was solved using partial fractions with complex coefficients.
jameson2
I'm trying to do a question that requires the expansion of the following using partial fractions:
$$f(z)=\frac{1}{(1+z^3)^2}$$.
The fact that the bottom is squared is throwing me off for some reason... I've factorized the bottom, but I'm not sure whether I should use the complex roots or not, or even if it's possible without using complex roots.
Any help would be appreciated.

Hey there,
The trick is to use identity 1+Z^3=(1+z)(1-Z+Z^2) and then just square this expression and do the partial fraction as usual.
Hope this helps)

$$\frac{1}{(1+z^3)^2}= \frac{1}{(z+1)^2(z^2-z+1)^2}$$
and can be written as "partial fractions" as
$$\frac{A}{z+1}+ \frac{B}{(z+1)^2}+ \frac{Cx+D}{z^2- z+1}+ \frac{Ex+F}{(z^2- z+1)^2}$$

There was an earlier question about the "principal part" or "Laurent series" for $1/(z^2+1)^2$ which I answered by reducing to partial fractions with complex coefficients. Is this related to that thread?

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## What is partial fraction expansion?

Partial fraction expansion is a method of breaking down a rational function into simpler fractions. It involves rewriting the function as a sum of simpler fractions with distinct denominators.

## What is the purpose of expanding 1/(1+z^3)^2 into partial fractions?

The purpose of expanding 1/(1+z^3)^2 into partial fractions is to simplify the function and make it easier to integrate or perform other calculations. It also allows us to better understand the behavior of the function and make predictions about its graph.

## How do you expand 1/(1+z^3)^2 into partial fractions?

To expand 1/(1+z^3)^2 into partial fractions, we first factor the denominator into its irreducible factors. Then, we set up a system of equations and solve for the unknown coefficients using algebraic manipulation and substitution.

## What are the restrictions on the values of z for partial fraction expansion of 1/(1+z^3)^2 to be valid?

The restrictions on the values of z for partial fraction expansion of 1/(1+z^3)^2 to be valid are that the denominator cannot equal zero, so z cannot be equal to the cube root of -1. Additionally, the function may have vertical asymptotes at values of z that make the numerator equal to zero.

## What are some real-world applications of partial fraction expansion?

Partial fraction expansion has several real-world applications in fields such as engineering, physics, and economics. It is commonly used in signal processing, control systems, and circuit analysis. It can also be used to solve differential equations and model complex systems.

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