课程名称︰复分析 课程性质︰数学系大三必修 课程教师︰蔡忠润 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2016/01/14 考试时限（分钟）：150 试题 : [Total: 37 points] In what follows, Ω is always assumed to be an open and connected proper subset of €. (1) [7 points] True/False questions, no justifications needed. ￣￣￣￣￣￣￣￣￣ (a) Given any three distinct complex numbers of unit length, ζ , ζ , and 1 2 ζ , define 3 z 1 f(z) = ∫ ───────────────── dw 0 2/3 ((w - ζ ) (w - ζ ) (w - ζ )) 1 2 3 for any z in the open unit disk. (Assume that the branch of 2/3-power is suitably chosen.) Then, we can find ζ , ζ , and ζ such that the 1 2 3 1 image of f(z) is a right triangle . (b) Suppose that f(z) is a function on ∂Ω, which is non-negative but not necessarily continuous. Let u(z) be the harmonic function on Ω produced 2 from the Perron's method with f(z). Then, u(z) ≧ 0 for any z ∈ Ω. (c) Suppose that ∂Ω is compact, and f(z) is a continuous function on ∂Ω. Then, there exists a harmonic function u(z) on Ω, which extends continuously to ∂Ω and which is equal to f(z) on ∂Ω. ︿ (d) For any α < -1, there exists an automorphism of € = € ∪ {∞}, f(z), such that f(0) = 0, f(1) = 1, and f(-1) = α. (e) Let Ω be the region enclosed by the heart curve, Ω = {(x, y) ∈ €| 2 2 3 2 3 (x + y - 1) - x y < 0}. Suppose that ψ(z) is an automorphism of Ω with ψ(0) = 0. Then, ψ(z) must be the identity map. 1 (f) Consider Ｆ = {f (z) =───} on €. It is normal, in the sense of n z + n n∈|N ︿ €. (g) There exists a non-constant entire function f(z), whose image does not contain the negative real axis. ∞ -2 (2) [5 points] Evaluate Σ n . Show your work. n=1 Hint: The zeros of sin(πz) are exactly Ｚ. Since ￣￣ sin(πz) = sin(πx) cosh(πy) + i cos(πx) sinh(πy) and cos(πz) = cos(πx) cosh(πy) - i sin(πx) sinh(πy), it is not hard to show that 1 ─|sin(πz)| ≦ |cos(πz)| ≦ 2|sin(πz)| 2 when |y| ≧ 1000. Another formula you might need is that ∞ 2 2 -1 π ∫ (a + s ) ds = ─. -∞ a (3) [2 + 5 points] Suppose that Ω is simply-connected. Fix a point z ∈ Ω. 0 Let Ｆ = {f(z): injective and analytic on Ω| f(z ) = 0, f'(z ) > 0, |f(z)| 0 0 < 1 for any z ∈ Ω}. (a) Why is Ｆ a normal family, in the sense of €? (b) Suppose that there exists a function g(z) ∈ Ｆ such that g'(z ) ≧ 0 f'(z ) for any f ∈ Ｆ. Prove that the image of g(z) is the (open) unit 0 disk. Namely, prove the surjectivity part of the Riemann mapping theorem. z - a 2 Hint: Given a ∈ D, let φ (z) = ───. Then, φ'(0) = 1 - |a| and ￣￣ a ─ a 1 - az 1 1 + |a| φ'(a) = ────. Another fact is that ────> 1 for any a ∈ D\{0}. a 2 ＿＿ 1 - |a| 2 √|a| (4) [5 points] Construct a harmonic function u(z) on the first quadrant, {z = x + iy ∈ €| x > 0 and y > 0} with the following property: ‧ u(z) extends continuously to the boundary except at the points 0 and 1; ‧ u(z) = 1 on the half-lines {y = 0, x > 1} and {x = 0, y > 0}; ‧ u(z) = 0 on the segment {0 < x < 1, y = 0}. ↑........... │........... u=1│........... │........... └──┴──→ 0 u=0 1 u=1 1 Hint: On the upper half space, ─ arg(z) is harmonic, and extends ￣￣ π continuously to the boundary except at the origin. Its value is 0 on the positive real axis, and is 1 on the negative real axis. (5) [6 points] Let Ｆ = {f(z): injective and analytic on Ω| f(z) ≠ 0 for any z ∈ Ω}. ︿ Prove that Ｆ is a normal family, in the sense of €. (6) [3 + 4 points] Let f(z) be an entire function. (a) Suppose that 0 is a pole of g(w) = f(1/w). Show that f(z) must be a polynomial. (b) Suppose that f(z) is not a polynomial. Then, given any w ∈ €, how 0 many roots does the equation f(z) = w have? Explain your reason. 0 Hint: You can think about what happens for f(z) = exp(z). ￣￣ ──── 1 That is to say, one angle of the triangle is π/2. 2 Namely, consider the supremum of certain subharmonic functions, u(z) = sup v(z). v∈Ｂ -- 2 2 1 ψxavier13540 给定一个二次元(|R )上的开集 G，设 f: G →|R ∈ C 。考虑一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，则 ╰x(0) = x° t ω(x°) 只能是一定点、一周期轨道或连接有限个 critical point 的连通路径，不会像三 次元一样可能出现混沌(chaos)。此即为 ODE 动力系统中的 Poincaré–Bendixson 定理。 --

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