1. 講者： 許永昌 老師 1

2. Contents Singular Pole Essential singularities Branch points Zero and root 2

3. Singular ( 請預讀 P372~P373) 3

4. Order of the pole and essential singularity 4

5. Zero and root 5

6. Branch Point ( 請預讀 P374~P376) Cauchy-Riemann Condition fx and f y are continuous. f ’(z) does exist f ’(z) does exist at z0 and its neighborhood. Analytic Taylor expansion Uniqueness theorem Natural boundary Analytic continuation n-sheeted surface (z-planes) Riemann Surface 6 Closed contour? It is obvious that sqrt(e -i  )  sqrt(e -i3  ) although e -i  =e -i3 . Branch point

7. Branch points (continue) 7 L1L1 L2L2 Branch point

8. Exercise 7.1.1 ( 請預讀 P374~P376) 8 1 or

9. The behavior at |z|   When we study the behavior of a function at |z|= , we usually do a reciprocal transformation z=1/  to study its behavior near  =0. The Laurent series for |z|   is 不是 (z-  ) n. The order of pole for |z|   is related to  –plane not z–plane. 9

10. Homework 7.1.1 7.1.2 7.1.3 10

11. Nouns 11