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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