Is it Possible to Construct a Laurent Series of Sqrt(z) About Zero?

In summary, the conversation discusses the possibility of constructing a Laurent series for sqrt(z) about zero by integrating over a keyhole contour and taking the limit as the radius of the inner circle approaches 0. However, there is disagreement about whether this is possible as sqrt(z) is a branch point at zero and Laurent series converge on an annulus. The idea of using a similar contour to find Laurent series for other functions at their branch points is also brought up.
  • #1
Ancient_Nomad
15
0
Hi,

My mathematics professor said that it is possible to construct a Laurent series of sqrt(z) about zero by integrating over a keyhole contour and then taking the limit R --> 0 where R radius of the inner circle. But I think he is mistaken. I don't understand how it is possible to have a Laurent series about zero, as it is a branch point.

Can someone please clarify this point, and tell me what the series is if such a series exists.

Also, then is it possible to have a laurent series for any function about its branch point by considering a similar contour.

Thanks.
 
Last edited:
Physics news on Phys.org
  • #2
I think you're right; Laurent series converge on an annulus, and square root cannot be defined* on an annulus about the origin.

Square root can be expressed by a (rather boring) Puiseux series, but I'm not sure how well that works complex analytically.


*: I mean in a continuous way, of course.
 

FAQ: Is it Possible to Construct a Laurent Series of Sqrt(z) About Zero?

1. What is a Laurent series?

A Laurent series is a mathematical representation of a complex-valued function in the form of an infinite sum of terms, including both positive and negative powers of a variable, typically denoted as z. It is used to extend the concept of a Taylor series to functions with singularities or poles.

2. How is a Laurent series different from a Taylor series?

A Laurent series includes both positive and negative powers of a variable, while a Taylor series only includes positive powers. This allows Laurent series to represent functions with poles or singularities, while Taylor series can only represent analytic functions.

3. What is the Laurent series of Sqrt(z)?

The Laurent series of Sqrt(z) is an infinite sum of terms, including both positive and negative powers of z, given by ∑n=-∞∞ an(z-c)n, where c is the center of the series and an is the coefficient of the nth term. In the case of Sqrt(z), the series is centered at c=0 and the coefficients are given by an = (-1)n(2n)! / (2nn!)(z-1)n.

4. What is the region of convergence for the Laurent series of Sqrt(z)?

The region of convergence for the Laurent series of Sqrt(z) is the set of all complex numbers z for which the series converges. In this case, the series converges for all complex numbers except for z=0, as this is the singularity of the function Sqrt(z).

5. How can the Laurent series of Sqrt(z) be used in calculations?

The Laurent series of Sqrt(z) can be used to approximate the value of the function at any point within its region of convergence. It can also be used for calculating integrals and derivatives of the function, as well as for finding the behavior of the function near its poles or singularities.

Similar threads

Question about Laurent Series Expansion
Replies
1
Views
2K
A Laurent series for algebraic functions
Replies
9
Views
2K
Confused about Laurent series of analytic functions
Replies
5
Views
5K
Finding the Coefficients in the Laurent Series of a Logarithm
Replies
3
Views
11K
What Are the Radii of Convergence for a Laurent Series Around a Singular Point?
Replies
5
Views
4K
Forming a Laurent Series for 4cos(z*pi) / (z-2i)
Replies
1
Views
2K
Can Degenerate Laurent Series Converge in Non-Standard Regions?
Replies
3
Views
2K
I What is the role of Laurent series in solving limits at infinity?
Replies
6
Views
2K
Expanding Laurent Series around Singularities
Replies
26
Views
2K
Elliptic functions, removable singularity, limits,
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top