课程名称︰工程数学 课程性质︰系必修 课程教师︰林沛群 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2008.05.12 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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