7.7 Differentiation and Integration of Fourier Series

Back

Theorem 7.3. The Differentiation Theorem If f is continuous and has

a piecewise continuous derivative f0 on [−_, _], then the Fourier series for

f0 is the series obtained by term by term differentiating of the Fourier Series

for f. If the second derivative f00 exists then the convergence is pointwise to

f0.

Theorem 7.4. The Integration Theorem For any f 2 PC[−_, _] with

Fourier series

a0 +

X1

j=1

(aj cos(jt) + bj sin(jt))

the function Z x

0

f(t) dt

has a series expansion

a0 x +

Xaj sin(jx)

j

−

bj cos(jx)

j

+ K

7.8. FUNCTIONS OF SEVERAL VARIABLES 153

where K is a constant. By putting x = 0 we find

K =

X1

1

bj

j

Using the series expansion for a0 x we obtain the Fourier series

X1

j=1

bj

j

+

X1

j=1

−bj cos(jx) +

􀀀

aj + (−1)k+1a0

_

sin(jx)

j

I shan’t prove either of these theorems, you will find them in KKOP.

They enable us to calculate some Fourier series relatively quickly and painlessly.

Навигация