MATEMATIK 4 KOMPLEKS FUNKTIONSTEORI MM 1.2 MM 1.1: Singulariteter og residuer Emner: Singulære punkter og nulpunkter Hævelig singularitet, pol, væsentlig singularitet Isoleret singularitet Residuer Regler til bestemmelse af et residuum i en pol

What should we learn today? How to classify singular points and zeros of a function and how singularities affect behavior of a function What is a residue? How with the help of residue to calculate complex integrals? Formulas for residues in case singularities are poles

Singularities and Zeros Definition. Funktion f(z) is singular (has a singularity) at a point z0 if f(z) is not analytic at z0, but every neighbourhood of z0 contains points at which f(z) is analytic. Definition. z0 is an isolated singularity if there exists a neighbourhood of z0 without further singularities of f(z). Example: tan z and tan(1/z)

Classification of isolated singularities Removable singularity. All bn =0. The function can be made analytic in z0 by assigning it a value . Example f(z)=sin(z)/z, z0 =0. Pole of m-th order. Only finitely many terms; all bn =0, n>m. Example 1: pole of the second order. Remark: The first order pole = simple pole. Essential singularity. Infinetely many terms. Example 2.

Classification of isolated singularities The classification of singularotoes is not just a formal matter The behavior of an analytic function in a neighborhood of an essential singularity and a pole is different.

Removable singularity Theorem. If an analytical function f(z) is bounded within a circle with some radius R (but without the center z0), then z0 is a removable singularity.

Pole Theorem. If f(z) goes to infinity for z  z0 , then f(z) has a pole in z0 .

Essential singularity Picard’s theorem If f(z) is analytic and has an isolated essential singularity at point z0, it takes on every value, with at most one exeprional value, in an arbitrararily small neighborhood of z0 .

Zeros of analytic function Definition. A zero has order m, if The zeros of an analytical function are isolated. Poles and zeros: Let f(z) be analytic at z0 and have a zero of m-th order. Then 1/f(z) has a pole of m-th order at z0 .

Analytic or singular at Infinity We work with extended complex plane and want to investigate the behavior of f(z) at infinity. Idea: study behavior of g(w)=f(1/w)=f(z) in a neighborhood of w=0. If g(w) has a pole at 0, the same has f(z) at infinity etc

Typeopgave Typical problem: Determine the location and kind of singularities and zeros in the extended complex plane. Examples:

L’hospital rule

Residue Integration Method We are interested in evaluation of integrals taken around a simple closed path C If f(z) is analytic  =0 by Cauchy’s integral theorem If f(z) has a singularity at z0 : Consider Laurent series that converges in a domain

Residue Integration Method The coefficient b1 is called the residue of f(z) at z0 and it is denoted by

Two formulas for residues at simple poles For a simple pole Another way to calculate the residue without the need of developing Laurent series:

Formula for the residue at a pole of m-th order

Multiple singularities inside the contour Theorem Let f(z) be analytic inside a simple closed path C and on C, except for finitely many singular points inside C. Then the integral of f(z) taken counterclockwise around C equals