Removable Singularities

Complex Analysis · Axiom Academy

LESSON Removable Singularities The mildest kind of singularity — a gap a function can be patched across, with a residue of exactly zero. Take . At z = 0 it's undefined — division by zero. But trace the graph in from either side and it heads for one clear value: The limit exists and is finite So define f(0)=1 — the gap is gone, f is now entire Because that limit L exists, we simply declare f(0) = L . The singularity is removable : it was never a true blow-up, only a point we'd forgotten to define. The patched function is analytic everywhere. How do you know in advance? Riemann's theorem says all of these are the same condition on an isolated singularity z_0 : is finite — and then f(z_0):=L . on some punctured neighborhood of z_0 . A nonzero (or infinite) product means a genuine blow-up, not a hole. Removable vs. pole, side by side For the product is , so the gap is removable. For the product is — the test fails, and z=0 is a pole. The single multiplier (z-z_0) tells the two apart. 3. No Principal Part, Zero Residue The deepest characterization is in the Laurent series about z_0 . A removable singularity has no negative powers — the whole principal part is gone: Every coefficient a_ -n with is zero. The would-be singular terms simply aren't there — what remains is an ordinary power series, and its constant term a_0 = L is the value we filled in. Worked check — at 0 : the numerator is , so dividing by z gives . No negative powers removable, with f(0)=1 .

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