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MAT 4 – Kompleks Funktionsteori 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

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MAT 4 – Kompleks Funktionsteori 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

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MAT 4 – Kompleks Funktionsteori Singularities and Zeros Definition. Funktion f(z) is singular (has a singularity) at a point z 0 if f(z) is not analytic at z 0, but every neighbourhood of z 0 contains points at which f(z) is analytic. Definition. z 0 is an isolated singularity if there exists a neighbourhood of z 0 without further singularities of f(z). Example: tan z and tan(1/z)

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MAT 4 – Kompleks Funktionsteori Classification of isolated singularities Removable singularity. All b n =0. The function can be made analytic in z 0 by assigning it a value. Example f(z)=sin(z)/z, z 0 =0. Pole of m-th order. Only finitely many terms; all b n =0, n>m. Example 1: pole of the second order. Remark: The first order pole = simple pole. Essential singularity. Infinetely many terms. Example 2.

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MAT 4 – Kompleks Funktionsteori 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.

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MAT 4 – Kompleks Funktionsteori Removable singularity Theorem. If an analytical function f(z) is bounded within a circle with some radius R (but without the center z 0 ), then z 0 is a removable singularity.

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MAT 4 – Kompleks Funktionsteori Pole Theorem. If f(z) goes to infinity for z z 0, then f(z) has a pole in z 0.

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MAT 4 – Kompleks Funktionsteori Essential singularity Picard’s theorem If f(z) is analytic and has an isolated essential singularity at point z 0, it takes on every value, with at most one exeprional value, in an arbitrararily small neighborhood of z 0.

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MAT 4 – Kompleks Funktionsteori 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 z 0 and have a zero of m-th order. Then 1/f(z) has a pole of m-th order at z 0.

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MAT 4 – Kompleks Funktionsteori 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

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MAT 4 – Kompleks Funktionsteori Typeopgave Typical problem: Determine the location and kind of singularities and zeros in the extended complex plane. Examples:

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MAT 4 – Kompleks Funktionsteori L’hospital rule

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MAT 4 – Kompleks Funktionsteori 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 z 0 : Consider Laurent series that converges in a domain

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MAT 4 – Kompleks Funktionsteori Residue Integration Method The coefficient b 1 is called the residue of f(z) at z 0 and it is denoted by

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MAT 4 – Kompleks Funktionsteori Two formulas for residues at simple poles For a simple pole Another way to calculate the residue without the need of developing Laurent series:

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MAT 4 – Kompleks Funktionsteori Formula for the residue at a pole of m-th order

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MAT 4 – Kompleks Funktionsteori 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

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