Beyond Taylor Series

Complex Analysis · Axiom Academy

A Taylor series can only reach the nearest singularity. To describe a function in the ring around one, we let the powers go negative. Every Taylor series hits a wall A Taylor series is wonderful machinery, but it is not universal. Centered at a point , it converges only inside a disk — and that disk can grow no farther than the nearest singularity of the function. Cross that boundary and the series falls apart, even in directions where the function itself is perfectly well behaved. So what do we do about the rest of the plane? First, watch the wall appear. We expand about the center . Its only singularity is the pole at . As more terms of the series switch on, the disk where they agree with the function fills outward — but it slams to a halt the instant it touches that pole. The radius can climb to and no farther. The series converges on and diverges on — the pole sets the wall. The series can feel the singularity coming Here is the rule that makes the wall predictable: a Taylor series about converges out to exactly the distance to the nearest singularity, . Drag the pole × around the plane. The disk where the series agrees with the function shrinks or grows to keep its rim pinned to the pole — never reaching past it, no matter the direction. Inside the disk, Taylor works perfectly. The open question is everything outside it — the ring around the singularity. Negative powers unlock the ring

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