課程名稱︰工程數學 下
課程性質︰必修
課程教師︰林沛群
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2010/5/5
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(55%) Consider functions f(x) and g(x) shown below
f(x)
^
(1)
＼ |
＼ |
_____＼|______1_> x (x與y的範圍皆在1與-1之間)
-1 |＼
| ＼
(-1) ＼
g(x)
|
(2)
|╲
| ╲
╲ (1) ╲
╲ |
__-1___＼|______1____> x
(a) (10%) Find the Fourier seriers of f(x) on the interval [-1 1]
(b) (10%) Use a convergence theory to determine the sum of the Fourier series
derived in (a), explain why, and plot this Fourier of f(x) vs. x
(c) (5%) Roughly sketch the first and tenth partial sum of the above series
(d) (10%) Derive the Fourier-Legendre series of f(x) on the interval [-1 1],
and roughly sketch its first and tenth partial sum.
(e) (10%) Find the phase angle form of the Fourier series of g(x) on the
interval [-1 1], and plot some points of the amplitude spectrum of
the function
(f) (10%) With extra definition
f(x) = 0 ︳x︳> 1,
find the Fourier integral of f(x), and plot this Fourier integral of
f(x) vs. x
2. (10%) If a is a nonzero real number, proof
1 ^ w
F[ f(at) ] (w) = ─── f (——)
｜a∣ a
3.(15%) Determine the Fourier transform of f(t)
f(t) = ─ cos(t) -k ≦ t < k
│
─ 0 t <-k or t≧k
4.(20%) Consider the Sturm-Liouville problem
y'' + λy = 0 y(0) = 0, y'(L) = 0
(a)(5%) Classify it is a regular, periodic, or singular problem; state the
revelant interval.
(b)(15%) Find the eigenvalues and corresponding eigenfunctions.
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