常见函数的傅里叶级数

∞ ♥ 2 2 0

0 0 ∑

24.4. c = f (x )e in π x /L dx = ♦1

(a + ib

n < 0 ♦ Definition of a Fourier Series

The Fourier series corresponding to a function f (x ) defined in the interval c ÷ x ÷ c + 2L L > 0 are constants, is defined as

where c and

24.1. a 0 + ∑ a

cos n π x + b sin

n π x 2

where

n n =1

L n L

♣a =

1 c +

2 L

n π x

f (x ) cos dx

24.2. ♠ n

L ⎰c

1 c +

2 L

L

n π x

♠b n = L ⎰c

f (x ) s in

L dx

If f (x ) and f '(x ) are piecewise continuous and f (x ) is defined by periodic extension of period 2L , i.e.,

f (x + 2L ) = f (x ), then the series converges to f (x ) if x is a point of continuity and to 1{ f (x + 0) + f (x - 0)} if x is a point of discontinuity.

Complex Form of Fourier Series

Assuming that the series 24.1 converges to f (x ), we have

24.3. f (x ) = ∑ c n e in π x /L

n =-∞

where

♣ 1 (a - ib )

n > 0 1 n 2L c +2 L - c ♠2 n n 2 - n - n ♠ ♥1 a n = 0

Parseval’s Identity

1

c +2 L

a 2 ∞ 24.5. { f (x )}2 dx = 0 + ∑

(a 2 + b 2 )

L ⎰c

n

n

n =1

Generalized Parseval Identity

24.6.

1

c +2 L

a c

∞

f (x )

g (x ) dx =

+ (a c + b d ) ∞

⎰ ) 2

L ⎰

c

2

n =1

n n

n n

where a n , b n and c n , d n are the Fourier coefficients corresponding to f (x ) and g (x ), respectively.

Special Fourier Series and Their Graphs

Fig. 24-1

Fig. 24-2

Fig. 24-3

Fig. 24-4

Fig. 24-5

Fig. 24-6 Fig. 24-7 Fig. 24-8 Fig. 24-9 Fig. 24-10

Fig. 24-11

Fig. 24-12 Miscellaneous Fourier Series