課程名稱︰應用數學三
課程性質︰物理系大二必修
課程教師︰何小剛
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰21/06/2011
考試時限（分鐘）：180分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Problem 1. (25 points)
a) Write down the expression for the Fourier transformed function f (x) and
f
Laplace transformed function f (s) of a function F(t).
l
b) Write down the expression for the inverse Fourier transform of the function
f (x), and inverse Laplace transform of the function f (s) in the Bromwich
f l
integral form.
∞
c) For a fourier series expansion f(x)=a_0/2+Σ(a_ncos(nπx/L)+b_nsin(nπx/L))
n=1
in the interval [-L,L], express a_n and b_n in terms of integration of product
of f(x), and cos(nπx/L) and sin(nπx/L).
∞ inπx/L
d) Show that f(x) can be written as f(x) = Σ c_n e and express c_n in
n=1
terms of a_n and b_n.
e) Write down the convolution formula which relates the functions F1(t) and
F2(t) and their Laplace transformed functions f1(s) and f2(s).
Problem 2. (25 points)
a) Obtain the Fourier series expansion for the function F(x) = x for -π< x <π
b) Obtain the Fourier transformed function f(t) of the function F(x) = x for
-π< x <π and 0 for |x| > π.
c) Obtain the inverse of Laplace tranform for the function f(s) = 1/(s-a)(s-b)
(s-c).
Problem 3. (25 points)
a) Obtain the Fourier transformed function f(ω) for the function F(x) = 1 for
|x| < 1 and F(x) = 0 for |x| > 1.
∞
b) Show that F(x) can be expressed as F(x) = (2/π)∫ dωsin(ω)cos(ωx)/ω.
0
c) Evaluate the integral in b) explicitly and show that for x = 1, F(x) = 1/2.
Problem 4. (25 points)
a) Obtain the Laplace transformed function x(s) for X(t) satisfying the driven
oscillator with damping equation: mX''(t) + bX'(t) + kX(t) = F(t) with initial
conditions X(0) = A (with A being a constant) and X'(0) = 0. (X'(t) = dX(t)/dt
2 2
and X''(t) = d X(t)/dt )
b) Using convolution formula to show that one can write the solution of X(t)
-bt/2m t -bz/2m
as: X(t) = Ae (cos(ω1t)+(b/2mω1)sin(ω1t))+(1/mω1)∫F(t-z)e sin(ω1z)dz
2 2 0
Here ω1 = (k/m)-(b/2m) .
c) Assuming that F(t) is equal to a for 0 < t < t0 and 0 else where. Obtain
the expression for X(t) by explicitly carrying out the integral in b).
2 2 ct
(Useful equations: the inverse of f(s) = d/((s-c) + d ) is equal to e sin(dt)
2 2 ct
and f(s) = (s-c)/((s-c) + d ) is equal ot e cos(dt). The Laplace transform
(n) (n) n n-1
for the nth derivative F of F is: L(F (t)) = s L(F(t)) - s F(0) - ...
(n-1)
- F (0)).
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※ 編輯: deepwoody 來自: 140.112.211.87 (06/27 00:20)