284 CHAPTER 12. SERIES AND TRANSFORMS

Example 12.3.3 Let f (x) = x for x ∈ [−π,π) and extend f to make it 2π periodic. Then

the Fourier coefficients of f are a0 = 0, ak =(−1)ki

k . Therefore,

12π

∫π

−π

te−ikt =ik

cosπk,∫ x

−π

tdt =12

x2− 12

π2 = lim

n→∞

n

∑k=−n,k ̸=0

(−1)k ik

∫ x

−π

eiktdt

= limn→∞

n

∑k=−n,k ̸=0

(−1)k ik

(sinxk

k+ i−cosxk+(−1)k

k

)For fun, let x = 0 and conclude − 1

2 π2 =

limn→∞

n

∑k=−n,k ̸=0

(−1)k ik

(i−1+(−1)k

k

)= lim

n→∞

n

∑k=−n,k ̸=0

(−1)k+1

k

(−1+(−1)k

k

)

= limn→∞

2n

∑k=1

(−1)k +(−1)k2 =

∞

∑k=1

−4

(2k−1)2

and so π2

8 = ∑∞k=1

1(2k−1)2

Of course it is not reasonable to suppose that you can differentiate a Fourier series termby term and get good results.

Consider the series for f (x) = 1 if x ∈ (0,π] and f (x) = −1 on (−π,0) with f (0) =0. In this case a0 = 0. ak =

12π

(∫π

0 e−iktdt−∫ 0−π

e−iktdt)= i

π

cosπk−1k so the Fourier se-

ries is ∑k ̸=0

((−1)k−1

πk

)ieikx.What happens if you differentiate it term by term? It gives

∑k ̸=0−(−1)k−1

πeikx which fails to converge anywhere because the kth term fails to converge

to 0. This is in spite of the fact that f has a derivative away from 0.However, it is possible to prove some theorems which let you differentiate a Fourier

series term by term. Here is one such theorem.

Theorem 12.3.4 Suppose for x ∈ [−π,π] f (x) =∫ x−π

f ′ (t)dt + f (−π) and f ′ (t)is piecewise continuous. Denoting by f the periodic extension of the above, then if f (x) =∑

∞k=−∞

akeikx it follows the Fourier series of f ′ is ∑∞k=−∞

akikeikx.

Proof: Since f ′ is piecewise continuous, 2π periodic it follows from Theorem 12.3.2

f (x)− f (−π) =∞

∑k=−∞

bk

(∫ x

−π

eiktdt)

where bk is the kth Fourier coefficient of f ′. Thus bk =1

2π

∫π

−πf ′ (t)e−iktdt. Breaking the

integral into pieces if necessary, and integrating these by parts yields finally

bk =1

2π

[f (t)e−ikt |π−π + ik

∫π

−π

f (t)e−iktdt]= ik

12π

∫π

−π

f (t)e−iktdt = ikak

where ak is the Fourier coefficient of f . Since f is periodic of period 2π, the boundary termvanishes. It follows the Fourier series for f ′ is ∑

∞k=−∞

ikakeikx as claimed.Note the conclusion of this theorem is only about the Fourier series of f ′. It does not say

the Fourier series of f ′ converges pointwise to f ′. However, if f ′ satisfies a Dini condition,then this will also occur. For example, if f ′ has a bounded derivative at every point, then bythe mean value theorem | f ′ (x)− f ′ (y)| ≤ K |x− y| and this is enough to show the Fourierseries converges to f ′ (x) thanks to Corollary 12.2.1.