# Principle part of Laurent series

• wam_mi
In summary, the principle part of the Laurent Expansion of f(z) about z=0 in the region 0 < mod z < 1 is given by the first two terms of the series expansion of e^z/(z+1) multiplied by 1/z^2. This is because e^z/(z+1) is nonsingular at z=0 and the other terms can be ignored.
wam_mi

## Homework Statement

Find the Principle part of the Laurent Expansion of f(z) about z=0 in the region
0 < mod z < 1, where f(z) = exp(z) / [(z^2)*(z+1)]

## Homework Equations

1/(1-z) = Summation (n = 0 to n = infinity) { z^n}

## The Attempt at a Solution

First, by using partial fraction,
I got f(z) = exp(z) {-1/z + 1/(z^2) + 1/(z+1)}

Then f(z) = exp (z) {1/ (-1+1+z) + 1/ (-1+1+(z^2)) + (1/(z+1) }

Since the question were only after the principle parts, so I ignore 1/(z+1) term

Basically I need to evalute
exp (z) { 1/ (-1+1+z) + 1/ (-1+1+(z^2)) }

Is this step right?

Then I tried to do the following,

and I got something like

exp (z) { - summation (1/(1+z))^(n+1) + summation (1/(1+z^2))^(n+1)}

But is this right?

Thanks a lot!

You are making this harder than it is. Your function is 1/z^2 times e^z/(z+1). e^z/(z+1) is nonsingular at z=0. All you need is the first two terms in the series expansion of e^z/(z+1) around z=0. (Why only the first two?). Then multiply that series by 1/z^2.

## 1. What is the principle part of a Laurent series?

The principle part of a Laurent series is the portion of the series that contains negative powers of the variable. It is the part of the series that is not analytic at the point of expansion.

## 2. How is the principle part of a Laurent series different from the regular Taylor series?

The principle part of a Laurent series includes both positive and negative powers of the variable, while a regular Taylor series only includes positive powers. This allows the Laurent series to represent functions that have poles or singularities at the point of expansion.

## 3. How is the residue calculated in a Laurent series?

The residue at a point of expansion in a Laurent series is calculated by taking the coefficient of the negative term with the lowest power of the variable. This coefficient is also known as the principal part coefficient.

## 4. Can a function have more than one Laurent series representation?

Yes, a function can have multiple Laurent series representations if it has multiple poles or singularities at the point of expansion. These different series will have different principle parts and residue values.

## 5. How is the Laurent series used in complex analysis?

The Laurent series is used in complex analysis to represent functions that are not analytic at a point. It allows for the study of functions with poles and singularities, and can be used to evaluate complex integrals and solve differential equations.

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