1. MAT 4 – Kompleks Funktionsteori MATEMATIK 4 KOMPLEKS FUNKTIONSTEORI MM 1.1 MM 1.1: Laurent rækker Emner: Taylor rækker Laurent rækker Eksempler på udvikling af Laurent rækker Singulære punkter og nulpunkter Hævelig singularitet, pol, væsentlig singularitet Isoleret singularitet

2. MAT 4 – Kompleks Funktionsteori KURSUSPLAN MATEMATIK 4 1. periode Tirsdag, 12:30-16:15 Kompleks Funktionsteori Induktion & Rekursion TKM Torsdag, 12:30-16:15 Tidsdiskrete systemer og sampling JoD Torsdag, 12:30-16:15 Lineær Algebra HEb 2. periode

3. MAT 4 – Kompleks Funktionsteori KURSUSPLAN MATEMATIK 4 kom.aau.dk/~tatiana/mat4 Her findes alt materialet til funktionsteori samlet: Opgaveløsninger, overheads, supplerende materiale Findes også på E4-hjemsiden. Kursuslitteratur: Del A Kompkeks Funktionsteori (2 mm) Kursuslitteratur: Finn Jensen & Sven Skyum Induktion og Rekursion Del B Induktion og Rekursion (3 mm) kom.aau.dk/~tatiana/mat4/IndukRekur.pdf

4. MAT 4 – Kompleks Funktionsteori What should we learn today? How to represent a function that is not analytical in singular points in form of a series We will call this series Laurent series How to classify singular points and zeros of a function and how singularities affect behavior of a function

5. MAT 4 – Kompleks Funktionsteori Reminder: Taylor series Taylor’s theorem: Let f(z) be analytic in a domain D, and let z 0 be any point in D. There exists precisely one Taylor series with center z 0 that represents f(z):

6. MAT 4 – Kompleks Funktionsteori Radius of convergence Taylor’s theorem: The representation as Taylor series is valid in the largest open disk with center z 0 in which f(z) is analytic. Cauchy-Hadamard formula:

7. MAT 4 – Kompleks Funktionsteori Laurent Series What to do if f(z) is not analytic in z 0 ? If f(z) is singular at z 0, we can not use a Taylor series. Instead, we will use a new kind of series that contains both positive integer powers and negative integer powers of z- z 0. Layrent’s theorem: Let f(z) be analytic in a domain containing two circles C 1 and C 2 with center z 0 and the ring between them. Then f(z) can be represented by the Laurent series

8. MAT 4 – Kompleks Funktionsteori Convergence region It is not enough to speak about radious of convergence. Laurent’s theorem: The Laurent series converges and represets f(z) in the enlarged open ring obtained from the given ring by continuosly increasing the outer circle and decreasing the inner circle until each of the two circles reaches a point where f(z) is singular. Special case: z 0 is the only singular point of f(z) inside C 2. Series is convergent in a disk Another way of determining region of convergence: it is an intersection of convergence regions of two parts of the series

9. MAT 4 – Kompleks Funktionsteori Uniqueness of Laurent series The Laurent series of a given analytic function f(z) in its region of convergence is unique. However, f(z) may have different laurent series in different rings with the same center.

10. MAT 4 – Kompleks Funktionsteori Typeopgave Typical problem: Find all Taylor and Laurent series of f(z) with center z 0 and determine the precise regions of convergence.

11. MAT 4 – Kompleks Funktionsteori Typeopgave Typical problem: Find all Taylor and Laurent series of f(z) with center z 0 and determine the precise regions of convergence. To find coefficients, we dont calculate the integrals. Instead, we use already known series.

12. MAT 4 – Kompleks Funktionsteori Examples

13. 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)

14. 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.

15. 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. Pole: a function can be made analutic if we multiply it with (z- z 0 ) m 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.

16. 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.

17. 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

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

19. MAT 4 – Kompleks Funktionsteori L’hospital rule