課程名稱︰複分析導論 課程性質︰系必修 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（年月日）︰2017/1/12 考試時限（分鐘）：120 試題 : Total points: 120 1. (20 points) Let α ∈ R - Z. Show that 　　　　　　 ｚ z Π ( 1 + －－－ ) exp ( -　－－－ ) n∈Z n + α n + α converges for all z ∈ C. 2. (20 points) Find the linear transformation which carries the line { z : Re z = Im z } into the circle | z + i | = 2, the point i into the origin, and the origin into i. 2 3. (20 points) Let Ω be a domain in C (identified with R ) and u(x,y) be a harmonic function defined on Ω. Show the following. a) If Ω is simply connected, then a conjugate harmonic function of u, defined on Ω, necessarily exists. b) Suppose Ω is not simply connected. Give an example, including a proof, in which a conjugate harmonic function defined on Ω does not exist. 2 4. (20 points) Let u(x,y) be a harmonic function on the entire R identified with C. Suppose that |u| is bounded, i.e. sup |u(x,y)| is finite. C Show that u is a constant function. (hint: try Schwarz formula and ordinarly Liouville theorem for bounded holomorphic functions). s+1 c+i∞ x ζ'(s) 5. (15 points) Let ψ (x) = ∫ --------( - ------ ) ds, x, c > 1. 1 c-i∞ s(s+1) ζ(s) By evaluating this integral by deforming the contours, show that ψ (x) 1 2 is asymptotically equal to x / 2 (as x → ∞). Note that the estimates on the norms of the Riemann zeta function ζ(s), ζ'(s) and 1/ζ(s) ε (all ≦ C |β| where β = Im s ≧ 1 or ≦ -1, 1 ≦ α ≡ Re s ≦ c, ε with any given ε > 0) can be used without proof. The fact that ζ(s) has no zeros along Re s = 1, can also be assumed without proof. -s/2 6. (15 points) Let ξ(s) = π Γ(s/2)ζ(s) where Γ(s) and ζ(s) are ∞ -x s-1 Gamma function ( = ∫ e x dx for Re s > 0) and Riemann zeta function 0 2 2 ∞ -πn x 1 ∞ - πn /x respectively. Assume the identity Σ e = ----- Σ e , x > 0. n=-∞ √x n=-∞ Assume also the interchangeability of the summation and integration. Show that the functional equation ξ(s) = ξ(1-s) holds, Re s > 1. 7. (10 points) Let f(z) be a holomorphic function defined on the domain 2 Ω = { 0 < |z| < 1+ε }, ε > 0 small. Suppose ∫∫ |f(z)| dx dy < ∞. Ω Show that a) f(z) can be extended as a holomorphic function across z = 0 (i.e. z = 0 is removable). (hint: try Laurent series and polar coordinates in the double integral.) b) Instead of f(z), consider a harmonic function u(x,y) satisfying similar conditions. In this case, does the similar conclusion hold for u? If not, give a counterexample. --

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