課程名稱︰複分析 課程性質︰數學系大三必修 課程教師︰蔡忠潤 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015/11/12 考試時限（分鐘）：150 試題 : [Total: 34 points] (1) [8 points] True/False questions, no justifications needed. ￣￣￣￣￣￣￣￣￣ ︿ (a) Let € = € ∪ {∞}. There exists a rational function q(z) such that -1 -1 ︿ ︿ #{q (0)} = 2 and #{q (∞)} = 0 (regard q(z) as a map from € to €, and count zeros with multiplicity). (b) There exists an analytic function f(z) defined on {z ∈ €| |z| < 2} (k) such that f (0) = 1 for any k ≧ 0. 1 (c) Consider the function f(z) = cosh─. There exist positive constants ε z and M so that |f(z)| > M for any 0 < |z| < ε. (d) Let D = {z ∈ €| |z| < 1}. Suppose that f: D → D is analytic with f(0) = 0. Then |f(z)| ≦ |z| for any z ∈ D. (e) There does not exist an entire function whose value on the positive real ∞ -t z-1 axis is ∫ e t dt. 0 (f) These does not exist an entire function whose value on n ∈ |N is (n-1)!. (g) These exists a meromorphic function whose pole is exactly n ∈ |N with j n exp(2 ) the singular part Σ ────. j=1 j (z-n) (h) Let B be the set of entire functions with f(0) = 1 and |f(z)| ≦ 999 when |z| = 100. For any n ∈ |N, there always exists f(z) ∈ B which has exactly n zeros in {z ∈ €| |z| < 50}. 3/4 (2) [5 points] Suppose that f(z) is an entire function obeying |f(z)| ≦ n when |z| = n ∈ |N. Prove that f(z) is a constant function. (Hint: There are (k) many ways to do it. You can show that f (0) = 0 for any k ∈ |N, or f(z ) 1 = f(z ) for any z , z ∈ €, etc.) 2 1 2 2015 304 12 4 3 (3) [5 points] How many roots does the equation z + 2z + 11z + z + z 2 + z + z + 1 = 0 have in {z ∈ €| |z| < 1}? Justify your answer. (4) [6 points] Evaluate the integral ax ∞ e ∫ ─── dx for 0 < a < 1. -∞ x 1 + e z (Hint: e has a period of 2πi.) (5) [6 = 2 + 4 points] Consider the function 2πz e - 1 f(z) =─────. z (a) Show that z = 0 is a removable singularity, and determine f(0). (b) Construct the Weierstrass product development of f(z), and find its genus. You shall briefly explain the convergence of your expression. You can simply invoke the theorem of Weierstrass, or derive the estimate by hand. (Hint: You may need the following formula: 2πz 2πe 1 1 1 ──── = π + ─+ Σ (─── + ─).) 2πz z n≠0 z - in in e - 1 (6) [4 points] Let f(z) be a non-constant entire function of finite order. Prove that the image of f(z) can miss at most one value in €. (Hint: You may assume f(z) misses α and β. What can you say about the entire function f(z) - α? Could it be possible that f(z) - α never equals to β - α?) -- 移居二次元(|R^2)的注意事項： 3. 如果你在從事random walk，往上下左右的 1. connectedness不保證pathwise connec- 的機率都是1/4，則你能回家的機率是1。 tedness。可能你跟你的幼馴染住很近， 4. 下面這個PDE是二次元上的波方程式 卻永遠沒辦法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不會出現 它的解不滿足Huygens' principle，因此 chaos，在預測事情上比較方便。 講話時會聽到自己的回音，很不方便。 --

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