作者t0444564 (艾利欧)
看板NTU-Exam
标题[试题] 109上 蔡宜洵 复分析导论 第五次小考
时间Fri Jan 15 15:27:49 2021
课程名称︰复分析导论
课程性质︰数学系大三必修
课程教师︰蔡宜洵
开课学院：理学院
开课系所︰数学系
考试日期︰2021年01月07日(四)
考试时限：13:10-13:50，共计40分钟
试题 :
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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