Kenneth Kuttler

658 CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIES

This proves the following.

Theorem 33.6.3 Let f denote the 2L periodic extension of the function f given on [−L,L)by

f (x) = f (−L)+∫ x

−Lf ′ (t)dt

and suppose f ′ is 2L periodic and piecewise continuous. Then for each x ∈ [−L,L] ,

f (x) = A0 +∞

∑n=1

Bn sin(nπx

∞

∑n=1

An cos(nπx

)where the Ak,Bk are the Fourier coefficients of f and the Fourier series for f ′ is

∞

∑n=1

Bnddx

sin(nπx

∞

∑n=1

Anddx

cos(nπx

)

33.7 Odd and Even ExtensionsOften, as in the above examples and in the applications which follow, the function you arefinding the Fourier series for is either even or odd. One way this often occurs is when thefunction of interest is defined on an interval [0,L] and it is only its values on this intervalwhich are of interest. Then you could consider either the even or the odd extension of thisfunction to [−L,L] and then extend it to be a 2L periodic function. For example, considerthe following pictures.

L−L

L−L L−L

The first of these is an even extension to [−L,L] and the second is an odd extensionto [−L,L]. In the first case where there is an even extension, the Fourier coefficients arebk = 0

a0 =1

∫ L

−Lf (x)dx =

∫ L

0f (x)dx

ak =1L

∫ L

−Lf (x)cos

(kπxL

)dx =

∫ L

0f (x)cos

(kπxL

)dx

In the second case where you are dealing with the odd extension, each ak = 0 and

bk =1L

∫ L

−Lf (x)sin

(kπxL

)dx =

∫ L

0f (x)sin

(kπxL

)dx

Example 33.7.1 Let f (x) = x on [0,1] . Find the Fourier series of its even extension.

658 CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIESThis proves the following.Theorem 33.6.3 Let f denote the 2L periodic extension of the function f given on |—L,L)byfa =s(-+ [if oarand suppose f’ is 2L periodic and piecewise continuous. Then for each x € |-L,L},f(x)= Aut Basin (™ L *)+ Yan cos ( a)where the Ay, By are the Fourier coefficients of f and the Fourier series for f' is— d nix — d nxBu gesin (TT) +L Angeos ()py ng, Sin +h np 0833.7 Odd and Even ExtensionsOften, as in the above examples and in the applications which follow, the function you arefinding the Fourier series for is either even or odd. One way this often occurs is when thefunction of interest is defined on an interval [0,Z] and it is only its values on this intervalwhich are of interest. Then you could consider either the even or the odd extension of thisfunction to [—L, L] and then extend it to be a 2L periodic function. For example, considerthe following pictures.The first of these is an even extension to [—L,L] and the second is an odd extensiono [—L,L]. In the first case where there is an even extension, the Fourier coefficients areb, =0L Law = sf fwar=7f feara = Ef reyeos(#) ae? [!royeos(#E) aIn the second case where you are dealing with the odd extension, each a, = 0 and1 pe k 2 pl kb= [ f(a)sin (=) ax== | f(x)sin (=) dxExample 33.7.1 Let f (x) =x on [0,1]. Find the Fourier series of its even extension.