数学物理方法第7章

Chapter 7 7.1 General Properties
Fourier series
Fourier Series
A Fourier series may be defined as an expansion of a function in a series of sines and cosines such as
sin 2 x sin 3x sin nx f ( x) x 2sin x (1) n1 . . 2 3 n
(7.15)
Figure 7.1 shows f(x) for the sum of 4, 6, and 10 terms of the series. Three features deserve comment. 1.There is a steady increase in the accuracy of the representation as the number of terms included is increased. 2.All the curves pass through the midpoint f ( x) 0 at x p
a0 f ( x) (an cosnx bn sin nx). 2 n1
(7.1)
The coefficients are related to the periodic function f(x) by definite integrals: Eq.(7.11) and (7.12) to be mentioned later on.

x (0,2p )
(7.20)
Both sides of this expansion diverge as x 0 and 2p
7.2 ADVANTAGES, USES OF FOURIER SERIES
Discontinuous Function
One of the advantages of a Fourier representation over some other representation, such as a Taylor series, is that it may represent a discontinuous function. An example id the sawtooth wave in the preceding section. Other examples are considered in Section 7.3 and in the exercises. Periodic Functions Related to this advantage is the usefulness of a Fourier series representing a periodic , that we expand it in functions . If f(x) has a period of 2p , perhaps it is only natural This guarantees that if a series of functions with period 2p , 2p 2 , 2p 3 our periodic f(x) is represented over one interval 0,2p or p , p the representation holds for all finite x.
r n cosnx 1 r n einx 1 r n einx . n 2 n1 n 2 n1 n n1

(7.18)
Now these power series may be identified as Maclaurin expansions of ln(1 z )
Sawtooth wave
Let us consider a sawtooth wave
0 x p x, f ( x) x 2p , p x 2p .
(7.14)
For convenience, we shall shift our interval from 0,2p to p , p . In this interval we have simply f(x)=x. Using Eqs.(7.11) and (7.12), we have
cos nx r n cos nx lim , r 1 n n n1 n1
(7.17)
absolutely convergent for |r|<1. Our procedure is to try forming power series by transforming the trigonometric function into exponential form:
an bn
1
p
1
pt cos ntdt 0,

p
p
pt sin ntdt p

p
2
p
0
t sin ntdt
p 2 p t cos nt 0 cos ntdt 0 pn 2 ( 1) n1 , n So, the expansion of f(x) reads
If we expand f(z) in a Laurent series(assuming f(z) is analytic),
f ( z)
On the unit circle
n
n d z n .

(7.4)
z ei and
i n
f ( z ) f (e )
(cosnx p
n 1
1 1

2p
0
f (t ) cos ntdt sin nx
2p
0
f (t ) sin ntdt)
0
p
n 1 0
2p
f (t ) cos n(t x)dt,
(7.13)
This equation offers one approach to the development of the Fourier integral and Fourier transforms.
a0 f ( x) (an cosnx bn sin nx). 2 n1
Express cos nx and sin nx in exponential form, we may rewrite Eq.(7.1) as
1 inx inx cos nx e e , 2
We can easily check the orthogonal relation for different values of the eigenvalue n by choosing the interval 0,2p

2p
0
p m,n , m 0, sin mxsin nxdx m 0, 0,
d e
n

in
.
(7.5)
The Laurent expansion on the unit circle has the same form as the complex Fourier series, which shows the equivalence between the two expansions. Since the Laurent series has the property of completeness, the Fourier series form a complete set. There is a significant limitation here. Laurent series cannot handle discontinuities such as a square wave or the sawtooth wave.
The Dirichlet conditions: (1) f(x) is a periodic function; (2) f(x) has only a finite number of finite discontinuities; (3) f(x) has only a finite number of extrem values, maxima and minima in the interval [0,2p]. Fourier series are named in honor of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series,
an
bn
p
1
1
2p
0
2p
f (t ) cos ntdt ,
(7.11)
(7.12)
p
0
f (t ) sin ntdt , n 0,1,2
Substituting them into Eq.(7.1), we write
1 f ( x) 2p 1 2p

2p
0 2p
f (t )dt f (t )dt
z re ix , re ix and
r n cos nx 1 ix ix ln( 1 re ) ln( 1 re ) n 2 n 1 ln (1 r ) 2r cos x
2



(7.19)


12
.
Letting r=1,
cos nx ln(2 2 cos x)1 2 n n 1 x ln(2 sin ), 2
(7.7)


2p
0
m 0, p m,n , cosmxcosnxdx 2p , m n 0,
sin m xcos nxdx 0
(7.8)
2p
0
for all integer m and n.
(7.9)
By use of these orthogonality, we are able to obtain the coefficients a0 f ( x) (an cosnx bn sin nx). 2 n1 multiplingcosmx, and thenintegralfrom0 to 2p
inx c e n


1 inx inx sinnx e e 2i


பைடு நூலகம்
f ( x)
in which
n
(7.2)
and
1 cn (an ibn ), 2 1 c n (an ibn ), 2 1 c0 a0 . 2
(7.3)
n 0,
Completeness
2p
a0 cos( m x ) f ( x ) dx 2 0
Similarly
2p
2p
cos(m x)dx (a cos(nx) cos(m x)dx b sin(nx) cos(m x)dx)
0 n 1 n n 0 0 2p 2p 2p

2p
2p
a0 sin(m x) f ( x)dx sin(m x)dx (an cos(nx) sin(m x)dx bn sin(nx) sin(m x)dx) 2 0 n 1 0 0 0
Figure 7.1 Fourier representation of sawtooth wave
Summation of Fourier Series Usually in this chapter we shall be concerned with finding the coefficients of the Fourier expansion of a known function. Occasionally, we may wish to reverse this process and determine the function represented by a given Fourier series. Consider the seriesn1 (1 n) cos nx , x (0,2p ). Since the series is only conditionally convergent (and diverges at x=0), we take
One way to show the completeness of the Fourier series is to transform the trigonometric Fourier series into exponential form and compare It with a Laurent series.