課程性質︰選修
開課系所︰化學系
2003. 6.20
1.State whether the given function is even or odd. Find the first three
nonvanishing terms of its Fourier series. Sketch the function and partial
sum with three terms. (5%*2=10%)
a)f(x)= x^2 / 2 (-π<x<π)
b)f(x)=｛x -π/2 < x < π/2
0
2.Find the first four nonvanishing terms of complex Fourier series of the
following functions: f(x) = x^2 (-π<x<π) (10%)
3.Show that the complex Fourier coefficients of an even function are real and
thost of an odd function are pure imaginary. (10%)
4.Find the steady-state oscillation of y''+cy'+y = r(t) with c>0 and
N
r(t) = Σ (a cos nt + b sin nt)
n=1 n n (10%)
2 ∞ 2 2 1 π 2
5.Using the Parseval's identity 2a + Σ (a +b ) = ─∫ f(x) dx , prove that
0 n=1 n n π -π
1 1 1 π^4
1 + ─ + ─ + ─ + ... = ─
3^4 5^4 7^4 96
(Hints:use f(x) = x if -π/2 < x < π/2
π-x if π/2 < x < 3π/2 ) (10%)
6.Find the Fourier transform of the following functions: (4%*4=16%)
a)f(x) = e^(-x) if x>0
0 if x<0
b)f(x) = x*e^(-x) if x>0
0 if x<0
c)f(x) = e^x if x>0
e^(-x) if x<0
d)f(x) = e^(-α*x^2) , α>0
7.a)Show that if f(x) has a Fourier transform, so do f(x-a), and
F{f(x-a)} = (e^(-iωa)*F{f(x)}
b)Using (a), obtain formula 1 in the following table, from formula 2.
^ ^
c)Show that if f(ω) is the Fourier transform of f(x), then f(ω-a) is the
Fourier transform of e^(iax)*f(x).
d)Using (c), oftain formula 3 from formula 1. (4%*4=16%)
┌─┬──────────┬──────^ ─────────────────┐
│ │ f(x) │ f(ω)=F(f) │
├─┼──────────┼────────────────────────┤
│1 │f(x)=1 if -b<x<b │ √(2/π) * (sin bω / ω) │
│ │ 0 otherwise │ │
├─┼──────────┼────────────────────────┤
│2 │f(x)=1 if b<x<c │(e^(-ibω) - e^(-icω))/(iω√2π) │
│ │ 0 otherwise │ │
├─┼──────────┼────────────────────────┤
│3 │f(x)=cosx if 0<x<a │(1/√2π)[(sina(1-ω))/(1-ω)+(sina(1+ω))/(1+ω)]
│ │ 0 otherwise │ │
└─┴──────────┴────────────────────────┘
8.Simultaneous Diagonalization
The Huckel matrix for cyclic polyene with N carbons is
α β β
β α β
β α β
H=[ … ]
β α β
β β α
verify by direct substitution that the eigenvectors of the Huckel matrix H
are given by the column vectors of Fourier matrix:
1 1 1 … 1
1 ω ω^2 … ω^(N-1)
1 ω^2 ω^4 … ω^(2(N-1))
C=[c1,c2,...,cN]=[… ] with
1 ω^(N-1) ω^(2(N-1)) … ω^((N-1)^2)
ω= e^(2πi/N) and ω^N = e^(2πi) = 1
Are these eigenvectors normalized? If not, what are the normalization
factor for each eigenvector? With these eigenvectors, it is straightforward
to find out the eigenvalues for the Huckel matrix H by matrix-vector
multiplication Hc =λ c
j j j (20%)
Total:102%
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