Define power series in complex analysis

How do you define a power series?
Power series is a sum of terms of the general form aₙ(x-a)ⁿ.
Whether the series converges or diverges, and the value it converges to, depend on the chosen x-value, which makes power series a function..
What is a function defined by a power series?
A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series.
This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a that converges to f(x) for every x ∈ V..
What is a well defined power series?
A formal power series F(x) is well-defined if it is an element in C[[x]], the ring of formal power series.
Here we have an infinite product F(x)=∏n≥1(1−xn)−μ/n. and in order to show that F(x)u220.
1. C[[x]] we have to assure that it converges in the standard topology of formal power series
What is the importance of power series method?
Power series is an important way to analyze functions in various settings.
When a function agrees with its power series it is said to be analytic.
Sometimes one specifies the interval over which the function is analytic.
The functions with power series representation take us beyond the polynomials..
What is the purpose of the power series?
In short, power series offer a way to calculate the values of functions that transcend addition, subtraction, multiplication, and division -- and they let us do that using only those four operations.
That gives us, among other things, a way to program machines to calculate values of functions like sin(x) and sqrt(x)..
A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series.
This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a that converges to f(x) for every x ∈ V.
Power series don't have to be centered at 0.
They can be centered at any point x=a, in which case the series will contain powers of (x−a) instead of powers of x.
• Definition: A (real or complex) function f(z) is called analytic at a point z0 if it has a power series. expansion that converges in some disk about this point (i.e., with ρ \x26gt; 0).
A singularity of a function is a point z0 at which the function is not analytic.

A power series is a series of functions ∑ fn where fn : z ↦→ anzn, (an) being a sequence of complex numbers. Depending on the cases, we will consider either the complex variable z, or the real variable x. Notations 1.2. For r ≥ 0, we will note ∆r = {z ∈ C | |z| < r}, Kr = {z ∈ C | |z| ≤ r} and Cr = {z ∈ C | |z| = r}.

Define power series in complex analysis

How do you define a power series?

What is a function defined by a power series?

What is a well defined power series?

What is the importance of power series method?

What is the purpose of the power series?