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 --

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