1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.7 Proof of Convergence 95

Use the fact that

∫∞

0

sin(t)

t dt=

π

2.

1.7 Proof of Convergence

In this section we prove the Fourier convergence theorem stated in Section 3.
Most of the proof requires nothing more than simple calculus, but there are
three technical points that we state here.

Lemma 1.For all N= 1 , 2 ,...,

1

π

∫π

−π

(

1

2

+

∑N

n= 1

cos(ny)

)

dy= 1. 

Lemma 2.For all N= 1 , 2 ,...,

1

2 +

∑N

n= 1

cos(ny)=

sin

(

(N+^12 )y

)

2sin(^12 y). 

Lemma 3.Ifφ(y)is sectionally continuous,−π<y<π,thenitsFouriercoef-
ficients tend to 0 with n:

nlim→∞^1

π

∫π

−π

φ(y)cos(ny)dy= 0 ,

nlim→∞

1

π

∫π

−π

φ(y)sin(ny)dy= 0. 

In Exercises 1 and 2 of this section, you are asked to verify Lemmas 1 and 2
(also see Miscellaneous Exercise 17 at the end of this chapter). Lemma 3 was
proved in Section 6.
The theorem we are going to prove is restated here for easy reference. Period
2 πis used for typographic convenience; we have seen that any other period
can be obtained by a simple change of variables.

Theorem.If f(x)is sectionally smooth and periodic with period 2 π, then the
Fourier series corresponding to f converges at every x, and the sum of the series is

a 0 +

∑∞

n= 1

ancos(nx)+bnsin(nx)=^12

(

f(x+)+f(x−)

)

. (1)