課程名稱︰工程數學下
課程性質︰必修
課程教師︰蕭浩明
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2020/4/29
考試時限（分鐘）：60
是否需發放獎勵金：是
試題 :
1.(a) Expand f(x) = ∣x∣, ∣x∣≦ π, in a Fourier series.
∞
(b) Use the result from (a) to find Σ 1/(2k-1)^2
n=1
2.Given the generating function for Legendre polynomials
∞
1/(1-2xt+t^2)^0.5 = Σ Pn(x) t^n
n=1
∞
express Σ Pn(cosθ) as a function of csc(0.5θ), 0＜θ＜2π.
n=0
sin(x), ∣x∣≦π
3.Use the Fourier integral to express f(x) = {
0 , ∣x∣＞π
∞
in terms of the integral ∫ g(α,x) dα.
0
∞ 1
4.If f(x) = Σ AnPn(x), derive its own Parseval's identity ∫ [f(x)]^2 dx
n=0 -1
in terms of An and n. Pn(x) are Legendre Polynomials.