課程名稱︰分析導論優一
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/01/13
考試時限（分鐘）：180
試題 :
1. (20%) Define
2
∞ -n t sin(nx)
f(t, x) = Σ e
n=1
Show that f(t, x) solves
2
∂f ∂ f
──- ── = 0 in (t, x) ∈ (0, ∞) ×(0, π)
∂t 2
∂x
and f(t, 0) = f(t, π) = 0 for all t > 0.
2. (20%) Determine all real values of x for which the following series
converges:
∞ 1 1 sin nx
Σ ( 1 + ─+ ... + ─) ───
n=1 2 n n
(Homework 8.26).
3. (20%) "Assume that both {f (x)} and {g (x)} converge uniformly on an interval
n n
I of |R. Then {f (x)g (x)} converges uniformly on I." Is this statement true?
n n
If your answer is "No", please modify the statement such that the new
statement is true. You will not get any credit if you consider trivial
modifications such as assuming that f (x) = a , g (x) = b or one of f (x) or
n n n n n
g (x) is a fixed constant for all n.
n
∞
4. A series Σ a is said to be Abel summable if there exists a function f(x) on
n=0 n
(-1, 1) such that
∞ n
f(x) = Σ a x
n=0 n
and lim f(x) exists.
x→1-
(a) (10%) Show that Cesaro summability implies Abel summability.
(b) (10%) Does the converse hold?
∞
5. Let {a } be a set of given real numbers.
n n=0
(a) (10%) Show that you can not always find a real analytic function f(x) in
(n)
I, an open interval of 0, such that f (0) = a .
n
∞ (n)
(b) (10%) However, you can always find a f(x) ∈ C (I) such that f (0) =
a . This is the so-called Borel's lemma. Here is how the proof goes.
n
∞
Assume that there exists a C (I) function φ(x) with φ(x) = 1 for x
near 0 and φ(x) = 0 for x ∈ I \ (-ε, ε) with ε > 0 and (-ε, ε) ⊆
I. Now define
∞
f(x) = Σ g (x),
n=0 n
where
n
x x
g (x) = φ(──) ─ a
n δ n! n
n
(n)
and δ → 0. It is easy to see that f (0) = a if we show that f ∈
n n
∞
C (I). To do so, we prove that we can choose appropriate δ such that
n
for 0 ≦ k ≦ n-1,
(k) -n
sup |g (x)| ≦ 2 .
x∈I n
Thus the series converges uniformly and can be differentiated term by
term infinitely many times. Hint: differentiate g (x) and treat x/δ as
n n
a new variable.
註：
1. 第三題的敘述，如果是對的就要給出證明，否則就要給出反例，並加上一些條件讓這個
敘述變成對的。
2. 第五題只要證明找得到那個 δ 就好了。
n