数学物理方法第7章

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 )
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

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.
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,
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
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
a0 f ( x) (an cosnx bn sin nfficients are related to the periodic function f(x) by definite integrals: Eq.(7.11) and (7.12) to be mentioned later on.
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.
(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.
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
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
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
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
inx c e n


1 inx inx sinnx e e 2i


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
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
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