Understanding Analytic Functions and Convergence

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In summary, an analytic function is a function that has a power series representation and converges absolutely within a certain radius of convergence. Convergence in this context refers to the limit of the sequence of partial sums in a series. This definition does not depend on the value of x, but rather on the function and the point x0. In the complex plane, a series is considered convergent if it is absolutely convergent. The definition also states that there must exist a real number R>0 where the series converges for all values of x within that radius. This definition can be found on Wikipedia's page about analytic functions.
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JamesGoh
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From my lecture notes I was given, the definiton of an analytic function was as follows:

A function f is analytic at xo if there exists a radius of convergence bigger than 0 such that f has a power series representation in x-xo which converges absolutely for [x-xo]<R

What I undestand is that for all x values, |x-xo| must be less than R (radius of convergence) in order for f to be analytic at xo.

Convergence in a general sense is when the sequence of partial sums in a series approaches a limit

Is my understanding of convergence and analytic functions correct ?
 
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JamesGoh said:
What I undestand is that for all x values, |x-xo| must be less than R (radius of convergence) in order for f to be analytic at xo.
What you're saying here would imply that the truth value ("true" or "false") of the statement "f is analytic at x0" depends on the value of some variable x. It certainly doesn't. It depends only on f and x0. (What you said is actually that if |x-x0|≥R, then f is not analytic at x0).

I'm a bit surprised that your definition says "converges absolutely". I don't think the word "absolutely" is supposed to be there. But then, in [itex]\mathbb C[/itex], a series is convergent if and only if it's absolutely convergent. So if you're talking about functions from [itex]\mathbb C[/itex] into [itex]\mathbb C[/itex], then it makes no difference if the word "absolutely" is included or not.

What the definition is saying is that there needs to exist a real number R>0 such that for all x with |x-x0|<R, there's a series [tex]\sum_{n=0}^\infty a_n \left( x-x_0 \right)^n[/tex] that's convergent and =f(x).

I like Wikipedia's definitions by the way. Link.
 

What does an analytic function mean?

An analytic function is a mathematical function that can be expressed as a power series of its variable. This means that the function can be written as an infinite sum of terms involving powers of the variable. In simpler terms, an analytic function is a function that can be represented by a series of algebraic operations.

What are the key characteristics of an analytic function?

Analytic functions have a few key characteristics that set them apart from other types of functions. These include being differentiable at every point within its domain, having a unique derivative at every point, and being continuous within its domain.

What is the difference between an analytic function and a non-analytic function?

The key difference between an analytic function and a non-analytic function lies in their ability to be represented by a power series. While analytic functions can be expressed in this way, non-analytic functions cannot. This means that non-analytic functions may exhibit discontinuities or have undefined derivatives at certain points within their domain.

Can an analytic function have multiple representations?

Yes, an analytic function can have multiple representations. This is due to the fact that a power series can be written in different forms, such as using different starting points or changing the order of terms. However, all of these representations are equivalent and will yield the same function when evaluated.

What is the importance of analytic functions in mathematics and science?

Analytic functions are important in many areas of mathematics and science. They are used to model various physical phenomena, such as in physics and engineering. They also play a crucial role in complex analysis, which has applications in fields like computer science, statistics, and economics. Additionally, analytic functions are used in the development of numerical methods for solving mathematical problems.

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