Discontinuities and Singularities

Complex Analysis · Axiom Academy

LESSON Discontinuities and Singularities Classifying the points where a complex function stops being analytic — and why the three isolated types behave so differently. 1. Removable — the hole you can fill The gentlest singularity. The limit exists and is finite , but f is either undefined at z_0 or defined to the "wrong" value. The graph has a single missing point — a hole — that you can patch by defining f(z_0) to be that limit. A hole at z=1 — but cancel the factor and it is just z+1 The limit is finite, so define f(1):=2 and continuity is restored For at z=0 , the limit is 1 (Taylor series, or L'Hôpital). The point is missing, not broken — set f(0)=1 and the function is continuous through the origin. 2. Pole — the controlled blow-up Here the function genuinely runs off to infinity: as . The model is . A pole is "tame" infinity — its blow-up has a definite order n , the smallest power of (z-z_0) that multiplies the singularity away. n is the smallest positive integer for which (z-z_0)^n f(z) has a finite, nonzero limit at z_0 . For at z=0 , we have . The order is 2 because z^2 f(z)=1 is finite and nonzero, while still blows up — so n=2 is the smallest power that tames it. The function does not approach any value, not even , as . No order of pole works; the singularity cannot be tamed by any power of (z-z_0) . The signature example is e^ 1/z at z=0 . Along on the real axis, ; along , . Two approaches, two different answers — so the limit simply does not exist in .

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