[試題] 109上 蔡宜洵 複分析導論 第四次小考
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︰2020年12月17日(四)
考試時限：15:10-15:40，共計30分鐘
試題 :
Complex Analysis
Quiz 4
Name: Student ID:
1. (20 points) Suppose that z is not in Z.
(i) (10 points) Show that
π^2 ∞ 1
-------------- = Σ ---------.
(sin(πz))^2 n=-∞ (z-n)^2
(ii) (10 points) Express
∞ 1
Σ -----------
n=1 z^2 - n^2
in a closed form.
2. (10 points) Let f be a function which is continuous on the closed unit disc
and analytic on the open disc. Assume that |f(z)| = 1 whenever |z| = 1.
Show that the function f can be extended to a meromorphic function, with at
most a finite number of poles in the whole plane. (Hint: Extend f to all C
______
_
by f(z) = 1/f(z) whenever |z| > 1, and argue as in Schwarz reflection
principle.)
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