[試題] 109上 蔡宜洵 複分析導論 第五次小考
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︰2021年01月07日(四)
考試時限：13:10-13:50，共計40分鐘
試題 :
Complex Analysis
1. Let ψ be Tschebychev's ψ-function defined by
log(x)
ψ(x) = Σ log(p) = Σ [ -------- ] log(p),
p^m≦x p≦x log(p)
where [u] denotes the largest integer less than or equal to u. If ψ(x)～x
as x→∞, the prove that π(x) ～ x/log(x) as x→∞. [Hint: You may need to
prpve two inequalities
log(x) log(x)
1 ≦ lim inf π(x) -------, lim sup π(x) ------- ≦ 1.
x→∞ x x→∞ x
For the second inequality, you may need to prove that
ψ(x) ≧ (π(x) - π(x^α)) log (x^α)
for any 0 < α < 1.]
2. Let f(z) be a non-constant elliptic function with period ω1,ω2, where
ω1/ω2 is not in R; and given z0 ∈ C, consider
P = {z0 + α1ω1 + α2ω2 | 0 ≦ α1, α2 ≦ 1}
be the fundamental parallelogram with no pole in ∂P. Prove tha
a.) f(z) has at least two poles in P.
b.) f(z) has same number of zeros and poles in P.
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※ 編輯: t0444564 (39.12.190.67 臺灣), 01/15/2021 15:28:40