I'm interested in the notion of convergence of a sequence of meromorphic functions by moving the poles. I'm taking a first course in (graduate) complex analysis, coming from a real analysis background.
Consider a convergent sequence $\{a_k: a_k \in \mathbb C \}_{k=0}^\infty$ and $a_k \to a^*$. Define
$$ f_k(z) := \frac{1}{(z - a_k)}, ~~~~~ f(z) = \frac{1}{z - a^*} $$ Intuitively, as $k\to\infty$, it seems obvious that $f_k \to f$ (as $a_k \to a^*$) and we should approach the function. Yet the standard choices of metric that characterize $\to$ ($L^1$, $L^2$ and their extensions into the complex plane) don't seem to handle singularities particularly nicely.
Near $a^*$, $f$ attains very large values, while $f_k$ at these points is potentially quite small, so the $L^\infty$/absolute value of the error is arbitrarily large. This seems to be the only hiccup to applying the Dominated Convergence Theorem.
My questions:
- Is there a standard notion of convergence for functions with poles or other types of singularities in complex plane?
- Is there a way to work this problem and apply the DCT?
- Can we find a way consider a broader class of rational functions where we "collide" poles (e.g. $f(z) = 1/(z - a^*)^2$ and $f_k = 1/((z - a^*)(z - a_k))$)