Hello! You’re exploring an interesting topic in complex analysis. Let's delve into it.

You're correct that analytic functions can indeed be expanded as infinite power series within their radius of convergence. For any function $f(z)$ that is analytic in a domain, it can be locally represented as a Taylor series:

$f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$

where $a_n$ are the coefficients obtained through differentiation.

However, concerning uniform convergence, let's clarify a couple of points:

1. Poles and Uniform Convergence: If $f(z)$ has poles within the domain, it cannot be represented by a Taylor series centered around that pole. Taylor series expansions are valid only in a disk around $z_0$ where the function is analytic and free of singularities (like poles).

   So, if we consider a sequence $f_n(z)$ that converges to $f(z)$ with poles, convergence can indeed fail to be uniform near these poles. This is because near a pole, the function values tend toward infinity, which disrupts the uniformity in convergence.

2. Uniform Convergence and Series Expansions: Uniform convergence is required if we want to interchange summation and limits. However, when dealing with sequences of analytic functions, if the convergence is not uniform, the limit function may lose some of the properties of the functions in the sequence. For example, a sequence of analytic functions that converges pointwise but not uniformly can converge to a function that is no longer analytic.

3. Convergence to Analytic Limits: When the sequence $f_n(z)$ converges uniformly on compact subsets (excluding poles), the limit function $f(z)$ will also be analytic within the same domain. This uniform convergence is possible only in regions where $f(z)$ is analytic without encountering poles.

In summary:

• Uniform convergence is indeed often impossible if the function $f(z)$ has poles within the considered region.
• Analytic functions with poles have regions where they cannot be expanded in a convergent Taylor series due to these singularities.
• If convergence to an analytic function is achieved without uniform convergence, it’s generally due to singularities (like poles) in the function.

Would you like a deeper dive into the proof structure for why Taylor series expansions require analyticity and uniform convergence?

Here are five related questions you might find interesting:

1. What is the impact of essential singularities on the convergence of a sequence of analytic functions?
2. How does Laurent series expansion accommodate functions with poles?
3. Can you explore the concept of meromorphic functions and their series representations?
4. What happens if a sequence of analytic functions converges uniformly on a larger domain with no poles?
5. How does the radius of convergence change when expanding around different points in a domain?

Tip: When studying complex functions with poles, consider the Laurent series, which can represent functions with isolated singularities in a given annular region.