MATEMATIK 4 KOMPLEKS FUNKTIONSTEORI MM 1.1

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

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

4. 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. Reminder: Taylor series • Taylor’s theorem: Let f(z) be analytic in a domain D, and let z0 be any point in D. There exists precisely one Taylor series with center z0 that represents f(z):

6. Radius of convergence • Taylor’s theorem: • The representation as Taylor series is valid in the largest open disk with center z0 in which f(z) is analytic. • Cauchy-Hadamard formula:

7. Laurent Series • What to do if f(z) is not analytic in z0 ? • If f(z) is singular at z0, 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- z0 . • Layrent’s theorem: • Let f(z) be analytic in a domain containing two circles C1 and C2 with center z0 and the ring between them. Then f(z) can be represented by the Laurent series

8. 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: z0 is the only singular point of f(z) inside C2. 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. 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. Typeopgave • Typical problem: Find all Taylor and Laurent series of f(z) with center z0 and determine the precise regions of convergence.

11. Typeopgave • Typical problem: Find all Taylor and Laurent series of f(z) with center z0 and determine the precise regions of convergence. • To find coefficients, we dont calculate the integrals. Instead, we use already known series.

12. Examples

13. 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)

14. 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.

15. 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- z0)m • 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 .

16. 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 .

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

19. L’hospital rule