??? ????? (Fourier Analysis)

1
??(?)

• ??? ????? (Fourier Analysis)
• ???? ?????? (Partial Differential Equations)
• ???? ????? (Complex Numbers and Functions)
• ???? ???? (Complex Integrations)
• ???? ???????? (Power Series, Taylor Series)
• ???? ????????? (Laurent Series, Residue
  Integration)
• ???? ???????????? (Complex Analysis Applied to
  Potential Theory)
• ????
• ???? 30
• ??? 35
• ??? 35

2
Chapter 10 ????? (Fourier Analysis)
?????????,????????????????????????.???????????????
?????????. ?????????????,?????????????.??????????
????,???????. ?????????,?????????????????????????
?,????????????????.
3
Chapter 10 ????? (Fourier Analysis)
?????(Periodic Functions)
????????? x ???????? p ????? f(x)??????
???? f(x)???????. p ??? f(x)?????(fundamental
period).
2p , 3p, 4p, np ?? f(x)???.
Q ?? f (ax) ??????
4
Chapter 10 ????? (Fourier Analysis)
?????(Periodic Functions)
?? f(x) ? g(x) ??????? p ?, ?
a, b ???
h(x)????????? p.
?? f(x) c ?? ???????,?????????.
????????????????
????????????
5
Chapter 10 ????? (Fourier Analysis)
?????(Trigonometric Series)
????????????? 2p????????
?????(trigonometric series)
?? a0, a1, a2, b1, b2, b3, .. ???
6
Chapter 10 ????? (Fourier Analysis)
?????(Fourier Series)
?????????(??? 2?)??????????,????????????(Fourier
series)
where
n 1,2,3,..
n 1,2,3,..
7
Chapter 10 ????? (Fourier Analysis)
??????????(Euler Formulas for the Fourier
Coefficients)
????? a0
8
Chapter 10 ????? (Fourier Analysis)
The Orthogonality Condition
for all integral m and n
9
Chapter 10 ????? (Fourier Analysis)
??????????(Euler Formulas for the Fourier
Coefficients)
????? an
10
Chapter 10 ????? (Fourier Analysis)
??????????(Euler Formulas for the Fourier
Coefficients)
????? bn
11
Chapter 10 ????? (Fourier Analysis)
???(Rectangular Wave)
12
Chapter 10 ????? (Fourier Analysis)
???(Rectangular Wave)
13
Chapter 10 ????? (Fourier Analysis)
???(Rectangular Wave)
Theorem 1
14
Chapter 10 ????? (Fourier Analysis)
Homework
Problem Set 10.2 8 16
15
Chapter 10 ????? (Fourier Analysis)
???? p 2L???(Functions of Any Period p 2L)
??? 2? ????????
??? 2L ????????
16
Chapter 10 ????? (Fourier Analysis)
???? p 2L???(Functions of Any Period p 2L)
17
Chapter 10 ????? (Fourier Analysis)
???? p 2L???(Functions of Any Period p 2L)
n 1
n 2, 3, .
n 2, 4, 6, .
n ????
18
Chapter 10 ????? (Fourier Analysis)
???? p 2L???(Functions of Any Period p 2L)
n 1
n 2, 3, .
19
Chapter 10 ????? (Fourier Analysis)
???????(Even and Odd Functions)
The Odd Function
f(x) is an odd function, e.g.
For all integer m and n
Fourier sine series
20
Chapter 10 ????? (Fourier Analysis)
???????(Even and Odd Functions)
The Even Function
f(x) is an Even function, e.g.
For all integer m and n
Fourier cosine series
21
Chapter 10 ????? (Fourier Analysis)
The General Function
22
Chapter 10 ????? (Fourier Analysis)
f(x)
Example 1 Square wave
2k
x
p
2p
3p
4p
-p
-2p
-3p
-4p
For n odd
For n even
f(x)-k is an odd function
23
Chapter 10 ????? (Fourier Analysis)
Example 2 Sawtooth wave
Theorem 2 --- Sum of functions
f1 f2 ??????? f1 ? f2 ??????????
24
Chapter 10 ????? (Fourier Analysis)
25
Chapter 10 ????? (Fourier Analysis)
Half-Range Expansions
f1 is an even periodic extension of f
f1 ? Fourier cosine series
f2 is an odd periodic extension of f
f2 ? Fourier sine series
26
Chapter 10 ????? (Fourier Analysis)
Half-Range Expansions
27
Chapter 10 ????? (Fourier Analysis)
Even periodic extension
28
Chapter 10 ????? (Fourier Analysis)
???????(Complex Fourier Series)
29
Chapter 10 ????? (Fourier Analysis)
???????(Complex Fourier Series)
30
Chapter 10 ????? (Fourier Analysis)
Example 1
f(x)ex, -?ltxlt?, f(x2 ?)f(x),???????????????
31
Chapter 10 ????? (Fourier Analysis)
????(Forced Oscillations)
32
Chapter 10 ????? (Fourier Analysis)
n 1,3,5,
33
Chapter 10 ????? (Fourier Analysis)
m 1g, c 0.02 g/s, k 25 g/s2
n 1,3,5,
??
????
34
Chapter 10 ????? (Fourier Analysis)
????????(Approximation by Trigonometric
Polynomials)
????(2?)?? f(x)?????????
??????N????,????????????!
35
Chapter 10 ????? (Fourier Analysis)
?
?
36
Chapter 10 ????? (Fourier Analysis)
??? ??????
???-? x ??,????F(x)???? f ??????,?F(N????)???
f ?????????,???? E ??????
?????(Bessel inequality)
?????
37
Chapter 10 ????? (Fourier Analysis)
?????(Fourier Integrals)
??????
?????
?????
?????
2Lgt2
38
Chapter 10 ????? (Fourier Analysis)
2L4
Amplitude spectrum
2L8
?2L 2k ?,???? 2k-1 -1???
2L16
?L ? ? , wn ? 0
39
Chapter 10 ????? (Fourier Analysis)
????? 2L ?????? fL(x)
?
? L ? ??????????
??????
40
Chapter 10 ????? (Fourier Analysis)
?????(Fourier Integrals)
41
Chapter 10 ????? (Fourier Analysis)
example
42
Chapter 10 ????? (Fourier Analysis)
?????????(Dirichlets discontinuous factor)
43
Chapter 10 ????? (Fourier Analysis)
x 0
????(sine integral)
44
Chapter 10 ????? (Fourier Analysis)
?L ? ? , oscillation? x ? 1
?????(Gibbs Phenomenon)
? w-wx-t
? wwxt
45
Chapter 10 ????? (Fourier Analysis)
???????(Fourier Cosine Integrals)
? f(x)????
B(w) 0
???????(Fourier Sine Integrals)
? f(x)????
A(w) 0
46
Chapter 10 ????? (Fourier Analysis)
example
?????????????????
???????
47
Chapter 10 ????? (Fourier Analysis)
???????
Laplace integral
48
Chapter 10 ????? (Fourier Analysis)
???????(Fourier Cosine Transforms)
???????
??
? f(x)????????
? ?????????
49
Chapter 10 ????? (Fourier Analysis)
???????(Fourier Sine Transforms)
???????
??
? f(x)????????
? ?????????
50
Chapter 10 ????? (Fourier Analysis)
example
? f(x) ? 0ltxlt? ???????k?,??????????
51
Chapter 10 ????? (Fourier Analysis)
????(?)????????
52
Chapter 10 ????? (Fourier Analysis)
??????????
?? f(x)? x ???????????,? f(x)????????????, ???x??
?f(x)?0,?
53
Chapter 10 ????? (Fourier Analysis)
??????????
54
Chapter 10 ????? (Fourier Analysis)
??????????
?? f(x)? x ???????????,? f(x)????????????, ???x??
?f(x)?0,?
55
Chapter 10 ????? (Fourier Analysis)
??????????
56
Chapter 10 ????? (Fourier Analysis)
??????????
????????
?w??
W????
???w????, ???F(w)
57
Chapter 10 ????? (Fourier Analysis)
??????????
If
58
Chapter 10 ????? (Fourier Analysis)
?????(Fourier Transforms)
? f ??????
? ???????
59
Chapter 10 ????? (Fourier Analysis)
Physical Interpretation Spectrum
?????????????????
????, ?? f(x)?????w?w?w?????
???????
60
Chapter 10 ????? (Fourier Analysis)
61
Chapter 10 ????? (Fourier Analysis)
????
???????,? y f(x)?????????????????,? ?????????
,????? cn ? ???,????? ?(discrete
spectrum)????(point spectrum),?????????
62
Chapter 10 ????? (Fourier Analysis)
??????????
63
Chapter 10 ????? (Fourier Analysis)
????????
?? f(x)? x ??????? ??? f(x)?0,??f(x)?x??????
??
64
Chapter 10 ????? (Fourier Analysis)
??????????(convolution theorem)
?? f ? g ?????
??????????(convolution theorem)
?? f(x)? g(x)? x ???????,????????
65
Chapter 10 ????? (Fourier Analysis)
? x p q, ? x p q
66
Chapter 10 ????? (Fourier Analysis)