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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）︰2008.05.12
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.(40%) Consider a function
-1 -4 < = x < 0
f(x) =
1 0 < = x < = 4
(a)(10%) Find the Fourier series of the function on the above interval
(b)(10%) Use a convergence theory to determine the sum of the Fourier
series derived in (a), explain why, and plot this F.s.
(c)(10%) Represent the Fourier series derived in (a) in the phase angle
form, and plot some points of the amplitude spectrum of the function.
(d)(10%) With extra definition
f(x)=0 lxl>4
find the Fourier integral of the function, and plot this F.i.
2.(15%) If a is a nonzero real number, proof
http://homepage.ntu.edu.tw/~b95502134/exam/1.bmp
3.(15%) Determine the Fourier transform of the function
(6e^(2it))/(t^2-4t+8) hint: F[1/(t^2+a^2)](w)=(π/a)*e^(-alwl)
4.(10%) Expand the following polynomial in a series of Legendre polynomials
2+3x-4x^2
5.(20%) Consider the Sturm-Liouville problem
y''+λy = 0 y(0) = 0 3y(1) + y'(1) = 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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