課程名稱︰工程數學
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）︰2017/4/10
考試時限（分鐘）：90
試題 :
∞
1. (15%) If ∫ f(x)cosωxdx = 1, 0 < ω ＜ π
0 0, π < ω < ∞
Find f(x)?
2. (15%) Given the Fourier tranform F[ 1/(x^2 + b^2) ] = (π/b) e^(-αb)
∞ y(u) 1
Solve for y(x) from the integral equation ∫─────── du = ────
-∞(x-u)^2 + a^2 x^2+b^2
3. (15%) Given f(x) = x^2, -π< x < π, and its corresponding Fourier constants
∞
are a0 = 2π^2/3 and an = 4(-1)^n/n^2, evaluate series Σ (1/n^4).
n=1
4. (15%) Given the Fourier sine transform Fs[e^-kx] = ω/(k^2 + ω^2), k > 0,
∞
evaluate tje integral ∫ ωsin2ω/(ω^2 + 9) dω.
0
5. (20%) Use the Fourier integral to express f(x) = h(1-|x|/a), |x| <= a
0 , |x| > a
∞
in terms of the integral ∫ g(cos(α, x)) dα.
0
6. (20%) (a) Find the Fourier transfrom of f(x) = 1, |x| < a
0, |x| > a
∞
(b) Use the result of (a) to evaluate ∫ sinαx cosαx/αdα.
-∞