課程名稱︰分析一
課程性質︰數學系選修，可抵分析導論一
課程教師︰齊震宇
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016.9.20
考試時限（分鐘）：50
試題 :
1. (In this question we adopt the convention that a‧∞＝∞ when a > 0 and that
(1/n)
0‧∞＝0.) For a ∈C (n∈N), limsup |a | is either a nonnegative number or
n n→∞ n
∞. Show that
∞ n (1/n)
(1) for any z∈C, if Σ a ‧z converges, then ( limsup |a | )|z|≦1, and
n=0 n n→∞ n
∞ n
(2) for any 0 < A < 1, Σ a ‧z converges uniformly (and absolutely) on
n=0 n
(1/n)
{z∈C | (limsup |a | )|z| < A}.
n→∞ n
∞ n
2. Show that for a ∈C (n∈N), if the power series f(z):= Σ a ‧z converges
n n=0 n
∞ n-1
for all z in an open set U⊆C, then f'(z):= Σ n‧a ‧z for z∈U.
n=0 n
2
/ it +t-i if -3≦t≦2; 1 n
3. Let γ(t):= Compute ∫ (－) dz for all n∈N.
\ (-5+5i)t+12-7i if 2≦t≦3. γ z