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1.3 Convergence of Fourier Series 73

12.Prove the orthogonality relations

∫a

0

sin

(nπx

a

)

sin

(mπx

a

)

dx=

{

0 , n=m,

a/ 2 , n=m,

∫a

0

cos

(

mπx

a

)

cos

(

nπx

a

)

dx=

{ 0 , n=m,

a/ 2 , n=m=0,

a, n=m=0.

13.Iff(x)is continuous on the interval 0<x<a, is its even periodic ex-

tension continuous? What about the odd periodic extension? Check espe-

cially atx=0and±a.

14.Justify Theorem 1 by considering the integral as a sum of signed areas. See

Fig. 4 for typical even and odd functions.

15.Justify or prove these statements.

a. Ifh(x)is an odd function, then|h(x)|is an even function.

b. Iff(x)is defined for all positive x,thenf(|x|)is an even func-

tion.

c. Iff(x)is defined for allxandg(x)is any even function, thenf(g(x))is

even.

d. Ifh(x)is an odd function,g(x)is even, andg(x)is defined for allx,

theng(h(x))is an even function.

1.3 Convergence of Fourier Series

Now we are ready to take up the second question of Section 1: Does the Fourier
series of a function actually represent that function? The wordrepresenthas
many interpretations, but for most practical purposes we really want to know
the answer to this question: If a value ofxis chosen, the numbers cos(nπx/a)
and sin(nπx/a)are computed for eachnand inserted into the Fourier series
off, and the sum of the series is calculated, is that sum equal to the functional
valuef(x)?
In this section we shall state, without proof, some theorems that answer the
question (a proof of the convergence theorem is given in Section 7). But first
we need a few definitions about limits and continuity.
The ordinary limit limx→x 0 f(x)can be rewritten as limh→ 0 f(x 0 +h).Here
hmay approach zero in any manner. But ifhis required to be positive only, we
getwhatiscalledtheright-hand limitoffatx 0 ,definedby

f(x 0 +)=h→lim 0 +f(x 0 +h)=hlim→ 0

h> 0

f(x 0 +h).