講者：許永昌老師 1. Contents Singular Pole Essential singularities Branch points Zero and root 2.

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講者：許永昌老師 1

Contents Singular Pole Essential singularities Branch points Zero and root 2

Singular ( 請預讀 P372~P373) 3

Order of the pole and essential singularity 4

Zero and root 5

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

Branch points (continue) 7 L1L1 L2L2 Branch point

Exercise ( 請預讀 P374~P376) 8 1 or

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

Homework

Nouns 11