The term analytic refers to a function on the complex plane that has certain qualities. For the purpose of this explanation analyticness of a function can be treated like the complex equivalent of smoothness.
A pole is a point on a function's domain that is undefined. The Taylor Series is a good way to begin discussion of the Laurent Series.
The Taylor Series is of the form:. This series describes any smooth function as a sum of an infinite power series. The area this series accurately describes the function is the region of convergence of the Taylor Series. This area is limited by where the series returns a finite answer. When describing the Laurent Series r is called the radius of convergence.
This sum allows a continuous function to be represented by a sum of a discrete set of values. The concepts of region of convergence and translating a discrete signal into a continuous signal relate directly to the Laurent Series. Using a background in the Taylor Series, the Laurent series can be viewed as the complex extension that allows for the existance of poles and can even give a basic description of those poles.
I only have a couple of example problems in the GRE practice book, and I failed to understand all of them. I recon that I am not getting the motivation of when to use the Taylor Series and when the Laurent. My book also did not generalize how to manipulate the Taylor Series to make it into a Laurent series, so can someone guide me to where I could learn this a little bit more with concrete examples and details or explain this to me?
I know I am asking a lot, but mathematics means the life to me and I want to do as good as possible on the GRE. Well, the taylor series only works when your function is holomorphic, the laurent series works still for isolated singularities.
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Create a free Team What is Teams? Learn more. Difference between the Laurent and Taylor Series. Ask Question. Asked 8 years, 7 months ago. Active 2 years ago. Laurent Series. We shall see later that Laurent series expansions are unique, and so this must be the Laurent series representation for e1 z. Found inside — Page defined and differentiable at all other points of some open disk with center at zo. In this case, the function f z is expansible in what Is it poisonous?
Dean,1 T. We shall see that many of the results we have already developed will allow us to draw important conclusions about such functions. Expanded about what point? Question on tangent lines and the center of an ellipse. As a Hindu, can I feed other people beef?
For a given function, expansion point, and annular region, the Laurent expansion is unique, and may be found by any legitimate mathematical method. JavaScript is disabled.
The Laurent series. Last edited: Jul 11, Here are a few examples of what. Designed for the undergraduate student with a calculus background but no prior experience with complex analysis, this text discusses the theory of the most relevant mathematical topics in a student-friendly manner.
Laurent's series may be used to express complex functions in cases where Taylor's series of expansion cannot be apllied.
Found inside — Page To achieve this, we will use local information given by the Laurent series expansion of solutions at some point c. How can I remove a stuck kitchen faucet cartridge? Not surprisingly we will derive these series from Cauchy's integral formula.
For example, given.
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