講者： 許永昌 老師 1

Contents Residue Theorem Evaluation of Definite Integrals Cauchy Principle values Some poles on the integral path. Pole expansion of Meromorphic Function Product Expansion of Entire function 2

Residue Theorem ( 請預讀 P378~P379) Laurent expansion Closed contour integration Cauchy’s integral theorem Residue Theorem 3 Besides, {residue at z   }   {residues in the finite z-plane}

Residue 4

Evaluation of Definite Integrals ( 請預讀 P379~P384) 4 types we will discussed here: Hint: (1) |z|=1, (2) cos  =(z+z -1 )/2, (3)sin  =(z-z -1 )/(2i) Related to Jordan’s Lemma. 5

Exercises ( 請預讀 P379~P384) Step 1: find the singular points. Step 2: find a suitable contour. Step 3: For branch point, we must consider the branch cut. For poles, find a -1 on each pole. Step 4:Residue theorem. 6 Code: 小心， (1) 確定  的範圍 (2) 一整圈為 2  。

Jordan’s Lemma ( 請預讀 P383) If (1) a>0, a  R, (2) lim |z|   f (z)=0, 0  arg(z)  , We get lim R   |I R |=0, Proof: 7

Cauchy Principle Value ( 請預讀 P384) 8 C

Pole Expansion of Meromorphic Functions ( 請預讀 P390~P391) 9

Pole Expansion of Meromorphic Functions (continue) Proof for |f(z)|<  R k case: The remainder for |f(z)|<  R k p+1 case: 10 記得，是 F(  ) 的 residue 。

Exercises Test the pole expansion for: Test them with the remainder to understand the meaning of |f(z)|<  R k p+1. 11

Example ( 請預讀 P391) Pole expansion of cotangent:  cot  z= Method I: Its pole is located at z=n, n  Z. We will find that Choose R k =k+0.5, we get (I did not test it) |f(z)|<  R k. Therefore, based on Mittag-Leffler theorem, we get Method II: The product expansion of 12

Product Expansion of Entire Functions ( 請預讀 P392~P394) An entire function with zeros at z 1,…, z n can be written as where g(z) is an entire function with no zero. Questions: How to find the number of zeros in a region? How to do this product expansion for an entire function? Key concept: 13

Product Expansion of Entire Functions (continue) How to find the number of zero points? How to do the expansion? From the pole expansion, if |F(z)/R k |< , we get 14 小心：此處要求

Example 15

Rouché’s theorem 16 Z-planew-plane 1

Homework 1, 2, 3, 4, 6, 9, 14, 16, 21, 22 17

Nouns 18