課程名稱︰複分析導論 課程性質︰數學系大三必修 課程教師︰蔡宜洵 開課學院：理學院 開課系所︰數學系 考試日期︰2021年01月14日(四) 考試時限：13:20-15:20，共計120分鐘 試題 : 　　　 Complex Analysis January 14, 2021 Total points: 115 Note. 1) Your arguments and calculation details must be clear and complete; the auxiliary theorems needed for any claim of your results must be explicitly indicated. Failing to do so may result in loss of partial or full points. 1. (20 points) Let Ω ⊂ C be an open subset and a sequence of functions fn ∈ Hol(Ω) be given. Suppose that fn converges uniformly to f ≠ 0 (不恆等於零) on every compact subset of Ω. Assume that fn(z) ≠ 0 for all z ∈ Ω and n ∈ N. Prove that f(z) ≠ 0 for any z∈Ω. 2. (20 points) Suppose that f is analytic on the open unit disk and continuous on the closed unit disk. Assume that f = 0 on an arc of the circle |z| = 1. Show that f(z) ≡ 0. (Hint: Schwarz reflection principle and identity theor- em) ∞ j 3. Let a sequnence of functions fk(z) = Σ a(k) z in the form of convergent j=0 j power series be given on D := {|z| < 3/2}. Assume that fk uniformly converges on D. i) (20 points) Given any ε > 0, there exist an N = N(ε)∈N such that for all m,n > N, | a(m)_j - a(n)_j | < ε holds for all j = 0,1,2, ... .(Hint: Cauchy integral formula). ii) (5 points) Prove or disprove a similar statement about | ja(m)_j - ja(n)_j | instead of | a(m)_j - a(n)_j |. 4. (20 points) Set the lattice L = { m+in | m,n∈Z}. Construct, with proof, an entire holomophic function f(z) such that f(z) has simple zeros precisely 1 at the points of L. (Hint: Σ ---------- < ∞ and infinite (m,n)≠(0,0) |m+in|^3 product representation) 5. (10 points) Write s = α + i β. Define 1 c+i∞ x^s f(x) = ------ ∫ ------- g(s) ds 2πi c-i∞ s(s+1) where c > 1, x > 0 and g(s) ∈ Hol({α > 0}\{1}) has a simple pole at s = 1 1/2 with residue 1. Assume further that with some C > 0, |g(s)| < C|β| 2 if |α|≧1/2 and |β|≧1. Show that f(x) ～ x / 2 as x → ∞. 6. (10 points) Let p(z) be the Weierstrass p-function. Find the first three nonzero terms of its Laurent expansion on 0 < |z| < r for some r > 0. What is the largest possible value of r? And why? Here 1 1 1 p(z) = ----- + Σ ( --------- - ----- ) z^2 0≠w∈L (z-w)^2 w^2 where L = { m+in | m,n∈Z }. 7. (10 points) Let S⊂R^3 be an orientable surface. Explain the meaning of isothermal coordinates on S. Assuming the existence of loal isothermal coordinates on S (compatible with the orientation of S), explain why S can admit a complex analytic structure. --

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