课程名称︰工程数学 下 课程性质︰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）︰2010/5/5 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.(55%) Consider functions f(x) and g(x) shown below f(x) ^ (1) ＼ | ＼ | _____＼|______1_> x (x与y的范围皆在1与-1之间) -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. --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 124.11.64.144