Laurent Series in Annulus

Complex Analysis · Axiom Academy

EXAMPLE Laurent Series in Different Annuli One function, the same center, three regions — and three different series. The singularities at split the plane into three regions, and the function has a different series valid in each one. Region I: — a Taylor series (no negative powers). Region II: — a true Laurent series. Region III: — a Laurent series in only. Nice work — you produced three legitimate series for the same function from the same center. Same center, many series: a function has one Taylor series, but it can have several Laurent series — one per annulus between consecutive singularities. The whole game is the geometric form: for each term , choose the expansion whose ratio has magnitude . Use where , and where . Negative powers signal what's enclosed: Region I has none (it's a Taylor series); each time a singularity falls inside your circle, its term flips to negative powers of . Worth noticing: in Region III the combined coefficient is , so the leading term is — the coefficient is . The region you're expanding in is part of the answer: change the annulus and you change the series.

This is the written version of the interactive lesson above. See the full Complex Analysis course.