This is from Stein's Complex Analysis textbook. From page 87, it reads,
We contend that the function $H=f-f_{\infty}-\sum_{k=1}^{n}f_k$ is
entire and bounded. Indeed, near the pole $z_k$ we subtracted the
principal part of $f$ so that the function $H$ has a removable
singularity there. Also, $H(1/z)$ is bounded for $z$ near 0 since we
subtracted the principal part of the pole at $\infty$. This proves our
contention, and by Liouville's theorem we conclude that $H$ is
constant.
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It said that $H$ has a removable singularity at some points, but then it concludes that $H$ is also entire. Singularity is defined as some point that is not defined but is defined in the neighborhood of that point, but then how do we conclude that it is entire if the function is not even defined at these singularity points?
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I somehow understand that if we subtract principal part from the function, we can differentiate at that point. But I don't know how to rigorously formalize this in the context of the function $H$, because it seems that we are subtracting a lot of $f_k$'s from only one function $f$.
Sorry if this is not an organized question, but can somebody give an insight? Thanks.
Best Answer
When a function has a removable singularity you can define the value at the singularity to make the function analytic there. It's an acceptable abuse of terminology to call the function $H$ both before and after patching it at the removable singularity.
You subtract one $f_k$ for each pole $z_k$. Those $f_k$ are analytic everywhere except at $z_k$.