Mastering Laurent Series Expansion: A Layman's Guide

[j-lee00]

Homework Statement

In attachment

The Attempt at a Solution

I break into partial fractions, then get stuck. Please help me in layman terms

f(z) = -1/3[3(z+1)] + 4/3[z+4]

Now I am stuck.

 

[gomunkul51]

Hi!

I will help you with a full list of steps for one of the parts:

$$\frac{4}{3(z + 4)} = \frac{4}{3(4(\frac{z}{4} + 1))} = \frac{1}{3(\frac{z}{4} + 1)} = \frac{1}{3(1 + \frac{z}{4})} = \frac{1}{3}\cdot \frac{1}{(1 + \frac{z}{4})} = \frac{1}{3}\cdot\frac{1}{(1-(-\frac{z}{4}))} = \frac{1}{3}\left ( 1 + (-\frac{z}{4}) + (-\frac{z}{4})^{2} + (-\frac{z}{4})^{3} +\cdots \right ) =$$

$$= \frac{1}{3}\left ( 1 - \frac{z}{4} + \frac{z^{2}}{4^{2}} - \frac{z^{3}}{4^{3}} +\cdots \right ) = \frac{1}{3} - \frac{z}{3\cdot 4} + \frac{z^{2}}{3\cdot 4^{2}} - \frac{z^{3}}{3\cdot 4^{3}} +\cdots$$

Remember that for a given geometric series to converge |x| has to be less than 1. which in your case means:

$$\left |\frac{z}{4} \right |<1$$I hope you will understand all the steps otherwise do ask.
P.S. I used the sum of the geometric series (which is also the Taylor and Laurent series): 1/(1 - x) = 1 + x + x^2 + x^3 + ...
|x|<1

 

[j-lee00]

Thanks for your help.

With the expansion that you made, how does this differ to the taylor expansion? or it the same?

 

[vela]

If a Laurent series contains only non-negative powers, it will be identical to the Taylor series for the function.

 

[j-lee00]

Are the negative powers of the Laurent expansion only as big as the largest pole?

 

[vela]

What do you mean by "largest pole"?

Note that

$$\frac{4}{3(z + 4)} = \frac{1}{3} \left( 1 - \frac{z}{4} + \frac{z^2}{4^2} - \frac{z^3}{4^3} + \cdots \right)$$

has a pole at z=-4 yet no negative-power terms. The Laurent series about z=0 of the function for |z|>4 is

$$\frac{4}{3(z + 4)} = \frac{4}{3z}\left( 1 - \frac{4}{z} + \frac{4^2}{z^2} - \frac{4^3}{z^3} + \cdots \right)$$

has an infinite number of negative-power terms.

 

[j-lee00]

What you wrote clarified my problem. tks