1. 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

2. 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

3. 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)

4. 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.

5. 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.

6. 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.

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

8. 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 .

9. 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 .

10. 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

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

12. L’hospital rule

13. 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

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

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

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

17. 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