推 duncan2002 :原PO流唱一哥 05/09 23:41
推 snow0112 :抖抖欸 05/11 18:51
課程名稱︰工程數學
課程性質︰系必修
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）︰2011/5/9
考試時限（分鐘）：100
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (50%) Figure below plots the functions f(x), g(x), and h(x) with defined
intervals shown in thick gray bars
f(x)
^
(1)
＼ |
＼ |
_____＼|______1_> x
-1 |＼
| ＼
(-1) ＼
g(x)
|
|
|
(1) ╱
| ╱
| ╱
________|╱_____(1)______> x
h(x)
|
╱ |(2)
╱ |
╱ |
| ╱
| ╱
________|╱___________> x
(-1) (1)
(a) (12%) Find and plot rhe Fourier series of f(x)
(b) (8%) Roughly sketch the first and tenth partial sums of the Fourier
sine series and Fourirt cosine series of g(x) vs.x
(c) (10%) Find the phase angle form of the Fourier series of h(x), and plot
some points of the amplitude spectrum of the function
(d) (10%) Derive the Fourier-Legendre series of f(x) and roughly sketch its
first and tenth partial sums.
(e) (10%) With extra definition f(x)=0 │x│>1, find and plot the Fourier
integral of f(x)
2. (10%) Proof time reversal property of the Fourier transform
F[f(-t)](w)=f(-w)
3. (15%) Determine and plot the Fourier sine integral of f(t) with k>0, and
derive Fourier sine transform of the same function
sin(t) 0≦t≦k
f(x)=
0 t＞k
4. (5%) Briefly desvribe when the Fourier series of f(x) equals to f(x)
5. (20%) Consider the Strum-Liouville problem
y"+λy=0 y(0)-2y'(0)=0 y'(1)=0
(a) (5%) Classify it is a regular, periodic, or singular problem;
state the relevant interval
(b) (15%) Find the rigenvalues and corresponding eigenfunctions
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