Laurent series / residual theorem

In summary, the conversation is discussing a solution that requires the use of the exponential Taylor series. The first step is to convert cos(1/z) into its complex number form, which is 0.5 (e^(i/z) + e^(-i/z)). The individual has attempted to manipulate 1/z and z^5 into the exponential Taylor series, but it has become messy. They are seeking ideas for a better approach.
  • #1
ex3e
1
0

Homework Statement



http://img243.imageshack.us/img243/4339/69855059.jpg

Homework Equations


i've heard that the solution requires the use of the exponential taylor series:
http://img31.imageshack.us/img31/6163/37267605.jpg

The Attempt at a Solution



i know that the first step is to convert cos(1/z) into it's complex number form which is:
cos(1/z) = 0.5 (e^(i/z) + e^(-i/z))
I've tried manipulating 1/z and z^5 into the exp.taylor series but it gets messy (might be doing it wrong). any ideas would be great!
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
I would say you don't have to convert the cosine to exponential form if you already know the Maclauren series of cosine.
 

FAQ: Laurent series / residual theorem

1. What is a Laurent series?

A Laurent series is a representation of a complex function as an infinite sum of powers of the complex variable, including negative powers. It is similar to a Taylor series, but is applicable to functions with singularities, such as poles, in addition to regular functions.

2. What is the difference between a Laurent series and a Taylor series?

The main difference between a Laurent series and a Taylor series is that a Laurent series includes both positive and negative powers of the complex variable, while a Taylor series only includes positive powers. In other words, a Taylor series is a special case of a Laurent series where all the negative powers are equal to zero.

3. How is the radius of convergence determined for a Laurent series?

The radius of convergence for a Laurent series is determined by the distance from the center of the series to the nearest singularity of the function. If the function has no singularities, then the radius of convergence is infinite.

4. What is the Residual Theorem used for?

The Residual Theorem is used to calculate the coefficients of a Laurent series. It states that the coefficient of the term with power n in the Laurent series is equal to the value of the function at the center of the series, multiplied by the nth derivative of the function at that point.

5. Can a Laurent series be used to approximate any function?

No, a Laurent series can only approximate functions that are analytic, meaning they have a derivative at every point in their domain. Functions with essential singularities, such as sin(1/z), cannot be approximated by a Laurent series.

Similar threads

Calculating Laurent series of complex functions
Replies
3
Views
740
Contour integral, taylor and residue theory question
Replies
3
Views
2K
What is the residue of f(z) = e^(-2/z^2) using a laurent series?
Replies
2
Views
1K
Finding residues with Laurent series.
Replies
1
Views
1K
Finding Laurent Series and Residues for Complex Functions
Replies
2
Views
2K
Expanding f(z) in a Laurent Series
Replies
3
Views
1K
Laurent expansion of ##ze^{1/z}##
Replies
12
Views
4K
Expanding f(z) in a Laurent Series for |z|>3
Replies
3
Views
1K
Laurent series of z^2sin(1/(z-1))
Replies
2
Views
3K
Singularities and Laurent series
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top