The Principal Part

Complex Analysis · Axiom Academy

Split a Laurent series in two — the negative-power half is the principal part, and on its own it tells you exactly what kind of singularity you have. 1. Splitting a Laurent Series in Two About an isolated singularity z_0 , a function has a Laurent series — a power series stretched to allow negative exponents: Analytic part — the ordinary power series ( ) Principal part — every negative-power term Cut the series at the exponent n=0 . Everything with is a genuine power series, analytic at z_0 . Everything with n < 0 blows up as — that piece is the principal part P(z) , and it carries all of the singular behavior. 2. Counting Terms Names the Singularity You do not need to know the function — only how many negative-power terms its principal part has. That count alone fixes the singularity type: Principal part is empty. The function is secretly bounded, e.g. ; define f(0)=1 and the singularity vanishes. The most-negative power present is (z-z_0)^ -m . For the order is m=3 . The negative powers never stop, so there is no lowest one. is the classic essential singularity. For a pole, the order m is the largest negative exponent that actually appears — the leftmost lit slot in the lattice. . Exactly one negative term, , so this is a pole of order 1 (a simple pole). 3. One Coefficient Runs Everything: the Residue

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