Annular Regions

Complex Analysis · Axiom Academy

LESSON The Home of a Laurent Series A Taylor series needs a disk — but a function with a pole needs a ring. Meet the annulus, the natural domain of the Laurent series. An annulus centered at z_0 is the region trapped between two concentric circles — a disk with a smaller disk punched out of its middle. Watch the outer disk grow to radius R , then the inner disk of radius r get cut away: what's left is the ring. The points whose distance from z_0 lies strictly between r and R 2. Two Series, Glued at a Ring A Laurent series has two halves. The analytic part (non-negative powers) behaves like an ordinary Taylor series — it converges inside a disk of radius R . The principal part (the negative powers 1/(z-z_0)^n ) is the new piece — it converges outside a disk of radius r . The series as a whole only converges where both do: their overlap, which is exactly the annulus. converges on the disk |z-z_0| < R . converges on the exterior |z-z_0| > r . The same function, two different series For about z_0=0 : on the disk |z| < 1 it's the pure Taylor series (no principal part). On the annulus |z| > 1 it's the pure principal part . One function, one center — but a different Laurent series in each region . 3. Singularities Cut the Plane Into Rings

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