課程名稱︰複變函數論
課程性質︰數學系大三必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2007/11/29
考試時限（分鐘）：10：20～12：10再延十分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
P.S.滿分150，記分方式未知
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Note that exam contains nine problem sets. Write down your answers on the
answer sheet. You should include all the necessary calculations and reasoning.
(1) (15 points) Compute ∫ x dz , where x is the real part of z.
|z|=1
(2) (15 points) Suppose f(z) is analytic in a region that contains a closed
____
curve γ.Show that ∫f(z)f'(z)dz is pure imaginary.
γ
e^z
(3) (15 points) Evaluate ∫ -------- dz.
|z|=1 z^2
(4) Suppose f is analytic in C and satisfies |f(z)| < |z|^2 for those z with
|z| > 1129.
(a) (10 points) Show that f has a pole of order at most 2 at ∞.
(b) (10 points) Show that there are A,B屬於C such that f(z) - Az^2 -Bz is
analytic at ∞.
(c) (10 points) Show that there are A,B,C屬於C such that f(z)=Az^2 + Bz + C
(5) (15 points) Suppose that f(z) is an analytic function for |z| < 3.
Show that if |f(z)|≦1 for |z|=2 , then |f'(z)| < 1 for |z| < 1.
(6) (15 points) Prove by use of Schwarz's Lemma that every one-to-one conformal
mapping of a disk onto another is given by a linear transformation.
∞ x^2
(7) (15 points) Evalute ∫ --------------- dx.
0 x^4 + 5x^2 + 6
2π dx
(8) (15 points) Evalute ∫ -------------.
0 5 + sin^2(x)
∞ x^(1/4)
(9) (15points) Evalute ∫ ------------dx.
0 1 + x^2
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※ 編輯: sxq 來自: 140.112.245.115 (12/01 22:17)