Laurent Series for f(z)=(1+2z)/(z^2+z^3)

In summary, a Laurent series is a mathematical representation of a complex function that uses both positive and negative powers of the variable z. It is calculated by finding singular points and choosing an annulus, and can be used to understand a function's behavior near its singular points and approximate its behavior in the chosen annulus. It can also be used to identify poles and residues of a function. The Laurent series for f(z)=(1+2z)/(z^2+z^3) differs from its Taylor series in that it includes both positive and negative powers of z, providing a more complete representation of the function's behavior near its singular points.
  • #1
opticaltempest
135
0
Would anyone be willing to check and comment on my work for finding the Laurent series of

[tex]f(z)=\frac{1+2z}{z^2+z^3}[/tex] ?

Page 1 - http://img23.imageshack.us/img23/7172/i0001.jpg"

Page 2 - http://img5.imageshack.us/img5/2140/i0002.jpg"

Page 3 - http://img15.imageshack.us/img15/2753/i0003.jpg"

I also displayed the pages below.

Page 1

http://img23.imageshack.us/img23/7172/i0001.jpg


Page 2

http://img5.imageshack.us/img5/2140/i0002.jpg


Page 3

http://img15.imageshack.us/img15/2753/i0003.jpg


Thanks in advance!
 
Last edited by a moderator:
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  • #2
It looks good to me!
 
  • #3
Thanks!
 

FAQ: Laurent Series for f(z)=(1+2z)/(z^2+z^3)

1. What is a Laurent series?

A Laurent series is a mathematical representation of a complex function using a combination of positive and negative powers of the variable z. It is used to describe the behavior of a function near a singular point, such as a pole or a branch point.

2. How do you calculate the Laurent series for a given function?

The Laurent series for a function can be calculated by first finding its singular points, then choosing an annulus (a region between two circles) that contains the singular point of interest. The series can then be expressed as a sum of a power series (with positive powers of z) and a negative power series (with negative powers of z).

3. What is the significance of the Laurent series for f(z)=(1+2z)/(z^2+z^3)?

The Laurent series for this function can be used to understand its behavior near its singular points, which in this case are at z=0 and z=-1. It can also be used to approximate the function in the chosen annulus, providing a way to analyze it and make predictions about its behavior.

4. Can the Laurent series for a function be used to find its poles and residues?

Yes, the Laurent series can be used to identify the poles and residues of a function. The poles are the values of z where the negative power series diverges, and the residues can be calculated using the coefficients of the negative power terms in the series.

5. How does the Laurent series for f(z)=(1+2z)/(z^2+z^3) differ from its Taylor series?

The Taylor series for this function would only include positive powers of z, as it is centered at z=0. The Laurent series, on the other hand, includes both positive and negative powers of z, as it is centered at z=-1. The Laurent series provides a more complete representation of the function, as it takes into account its behavior near both of its singular points.

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