課程名稱︰複變函數論
課程性質︰數學系大三必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010.6.22
考試時限（分鐘）：三小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題：
There are 7 problems, each is of 15 pts except the first one which is of 10 pts
1.show that
∞ 1-cosx
∫----------- dx = π/2
0 x^2
(課本第二章例題)
2.
(a) For 0<a<1, show that
∞ v^(a-1)
∫------------dv = π/sinπa
0 1+v
(b)Use it to derive the functional equation for Γ
Γ(s)Γ(1-s)=π/sinπs for all s on C
3.
(a)Prove that for Res>1
∞ x^(s-1)
ζ(s)=(1/Γ(s))∫----------- dx
0 exp(x)-1
(b)Show that ζ(s) is continuable to s 屬於 C with only singularity a simple
pole at s = 1
4. ∞
Letφ(x):= Σ log(p) = Σ Λ(x) and φ1(x) = ∫φ(u)du
p^m<x n<=x 1
c+i∞ x^(s+1) -ζ(s)
Show that φ1(x) = (1/2πi)∫ ----------(--------)ds for any c>1
c-i∞ s(s+1) ζ(s)
5.State and prove the schwartz lemma for f:D → D with f(0)=0. Use it to
determine the group Aut(D) of biholomorphic maps of f.
6.Describe the conformal mapping
z dζ
f(z) = ∫-------------- z 屬於 H
0 (1-ζ^2)^1/2
in details.Describe also the inverse map of f
7.
(a) show that p'^2 = 4(p-e1)(p-e2)(p-e3) where ei:=p(wi/2)
(b) Let Ω 包含於 C be a simply connected domain not containing ei. Show that
w0 ds
I(w):=∫----------------- w0,w 屬於 Ω
w (4s^3-g2s-g3)^1/2
defines a inverse of p(z+a) for some a
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.4.200