Laurent series Flashcards
Annulus
An annulus is a set of the form { z e (C : R1 < |z-zo|< R2}
Laurents Theorem
Suppose that A is the annulus { z e (C : R1 < | z- zo| < R2}, that f e H(A) and that ^r is the circle {z e (C : | z-zo| = r}, traversed in the anticlockwise direction where R1< r < R2. Then
f(w) = SUM (n=-inf, inf) cn (w-zo)^n for all w e A.
where cn = 1/ (2Pii) Integral (over ^r) f(z) / (z-zo)^(n+1) dz
Laurent series
A series that converges in an annulus
Isolated singularity
A function f has an isolated singularity at zo in (C if f is holomorphic in the punctured ball B’(zo , r) for some r e |R+ and f is not differentiable at zo, possibly because f(zo) is not defined.
Nonisolated singularity
Every neighbourhood of the singularity contain other points where f is not differentiable
Laurent series of a function with an isolated singularity
If f has an isolated singularity at zo, there exists a positive number r such that the function is differentiable at any point of B’(zo , r) = { z e (C : 0 < |z-zo|
cn = 0
If cn = 0 for all n e |N, then f is identically 0 in B’(zo, r), so f(zo) is zero and f e H(B(zo,r)))
Essential singularity
We say that f has an essential singularity if there are infinitely many n e Z- such that cn =/0.
Removable singularity
We say that f has a removable singularity at zo if there are no n e Z- such that cn =/0
Pole
We say that f has a pole if there exists M e Z- such that Cm =/0 and cn=0 for all n
Zero of order M
If zo is an removable singularity. We say that f has a zero of order M at zo if there exists M e Z+ such that Cm =/0 and cn=0 for all n
Pole of order N
If f has a pole, we say that f has a pole of order N at zo if M = -N where N>0
Simple poles
Poles of order 1
