課程名稱︰工程數學下
課程性質︰機械系大二必修
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）︰104/4/8
考試時限（分鐘）：70 min
試題 :
1.(a) Find the Fourier cosine transform of f(x)=e^(-mx), m > 0. (10%)
╭∞ cos(pv)
(b) Use the result from (a) to show that│ ————— dv = ? (10%)
╯0 v^2+β^2
1
(c) Use the result from (b) to find the Fourier transform of ────. (10%)
x^2+ b^2
(d) Use the result from (c) to solve for y(x) the integral equation: (20%)
╭ ∞ y(u) 1
│ ─────── du = ──── 0 < a < b
╯-∞ (x-u)^2+ a^2 x^2+ b^2
Hint: Convolution
2.(a) Expand f(x) = x, 0 < x < 2, in a half-range sine series and write the
first three terms. (15%)
(b) Find a Fourier series for f(x) = x^2, 0 < x < 2, by integrating the
series of (a) on both sides of the equation from 0 to x directly. (15%)
∞ (-1)^(n-1)
(c) Use the result from (b) to evaluate the series Σ────── = ? (20%)
n=1 n^2
Hint: Consider a0(non-cosine terms).