Holomorphic at infinity (definition)
I struggle quite a bit with the usage of $\infty$ in complex analysis. In some cases, I can translate a definition involving infinity to equivalent statements using limits, or in the case of continuity I just make use of the topology defined on the Riemann sphere.
However, what I don't understand is why the notion of differentiability is defined at the point infinity. A function $f(z)$ is said to be holomorphic at $\infty$ if $f(1/z)$ is holomorphic at $z=0$. Same can be said about singularities. I just don't see why we would do this. For instance, when I'm asked to determine singularities, it often forget to check the point $\infty$, because the definition feels so arbitrary: yea, let's check this random point which is actually a limit, and then somehow that tells us something?
My source of confusion lies in the fact that I don't see why being holomorphic at $\infty$ tells us anything (well, I guess, besides that our function is bounded). I understand that being holomorphic around some complex number $a$ is useful, as we can then write our function as a power series around $a$, of in the case of isolated singularities, we can work with Laurent expansion. But we don't have these things when it comes down to $\infty$ (except when we switch to $f(1/z)$, but that's a whole different function now, no?)
I hope someone understands my confusion and could shed some light on this issue.
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