Does there exist complex function which is analytic in all $\Bbb{C}$ except for 5 points?
I would say it exist. $$f(z)=\frac{1}{z(z-1)(z-2)(z-3)(z-4)}$$ But how should I prove that it is analytic everywhere except for $z=0,1,2,3,4$?
2026-03-24 22:06:27.1774389987
Does there exist complex function which is analytic in all $\Bbb{C}$ except for 5 points?
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The ordinary quotient rule and product rule show that this is differentiable everywhere except at those five points. If "analytic" is taken to mean locally equal to the limit of its Taylor series, then a standard theorem says it's analytic at each point in $\mathbb C$ at which it's differentiable everywhere in some open neighborhood of that point.