L10A
第一次期中考第1,2,3題
第一次期中考第1,2,3題
第1題: Suppose that f is holomorphic in a domain and that, at every point,
either f=0 or f'=0. Show that f is a constant.
either f=0 or f'=0. Show that f is a constant.
第2題: Evaluate the given line integral. See 3:50.
第3題: Evaluate the given principal value integral. See 11:35.
0:00 第1題
3:11 第2題
11:08 第3題
L10B
第一次期中考第4,5,6題
第一次期中考第4,5,6題
這裡U代表複平面上的單位開圓盤
第4題: Let f be holomorphic on U. Let J={exp(it) : t in [0,pi/6)}.
Suppose f is continuous up to J and f(z)=0 on J. Is f identically equal to zero on U?
Prove it or give a counterexample.
Suppose f is continuous up to J and f(z)=0 on J. Is f identically equal to zero on U?
Prove it or give a counterexample.
第5題: Let f be a nonconstant entire function on C. Prove that the image of f is dense in C.
(No credit will be granted if you directly apply Picard's little theorem.)
(No credit will be granted if you directly apply Picard's little theorem.)
第6題: Let f be holomorphic on U and continuous on the closure of U.
Suppose that f is real on the boundary of U. Prove that f is a constant function on U.
Suppose that f is real on the boundary of U. Prove that f is a constant function on U.
0:00 第4題 (常見錯誤)
6:07 第4題 (法一)
7:45 第4題 (法二)
10:38 第5題
11:45 第6題 (法一)
13:01 第6題 (法二)
16:39 第6題 (法三)
20:11 第6題 (法四)
L10C
第一次期中考第7題、H. Cartan定理、Schwarz引理
第一次期中考第7題、H. Cartan定理、Schwarz引理
這裡U代表複平面上的單位開圓盤
第7題: Let f be a holomorphic function from U to U
satisfying f(0)=0 and f’(0)=1. Prove that f(z)=z.
satisfying f(0)=0 and f’(0)=1. Prove that f(z)=z.
0:00 H. Cartan's Theorem 亨利．嘉當定理
5:49 Proof 證明
10:37 Schwarz's Lemma 施瓦茨引理
14:17 Proof 證明
第10講 檢討第一次期中考、H. Cartan定理、Schwarz引理版書.pdf
