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10.4: EVEN AND ODD FUNCTIONS

KIAM HEONG KWA

1. The Fourier Cosine and Sine Series

It is often desired to expand in a Fourier series of period 2L a func-tion f(x) originally defined only on either one of the intervals (0, L),(0, L], [0, L), and [0, L]. One way this can be done is the following.We first extend f(x) to a new function onto either one of the intervals(L, L), (L, L], [L, L), and [L, L]. Then the new function is ex-tended periodically onto the real line R. Of particular simplicity andapplicability to us later are the following two extensions:

The even extension of f(x). Let

(1.1) g(x) =

f(x) if 0 x L,

f(x) if L < x < 0.

Then we require that g(x + 2L) = g(x) for all x R. Thefunction g(x) is called the even extension of f(x).

The odd extension of f(x). Let

(1.2) h(x) =

f(x) if 0 < x < L,

0 if x = 0, L,

f(x) if L < x < 0.

Then we require that h(x + 2L) = h(x) for all x R. Thefunction h(x) is called the odd extension of f(x).

See the appendix for some important properties of even and odd func-tions. The Fourier series of an extension of f(x) converges to f(x)almost everywhere in the sense that the series converges to f(x) in theoriginal interval except possibly at a finite number of points.

Let g(x) be the even extension of f(x). Let {an}

n=0and {bn}

n=1be

the Fourier coefficients of g(x). Then

an =1

L

L

L

g(x)cosnx

Ldx =

2

L

L

0

f(x)cosnx

Ldx, n = 0, 1, 2, ,

Date: February 12, 2011.

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2 KIAM HEONG KWA

and

bn = 1L

LL

g(x)sin nxL

dx = 0, n = 1, 2, 3, ,

because g(x)cosnx

Lis even and g(x)sin

nx

Lis odd. We define the

Fourier series F[g](x) of g(x) to be the Fourier cosine series Fc[f](x) off, i.e., we set Fc[f](x) = F[g](x). Explicitly,

Fc[f](x) =a0

2+

n=1

an cosnx

L,(1.3)

where

an =2

L

L

0

f(x)cosnx

Ldx, n = 0, 1, 2, .

Likewise, if h(x) is the odd extension of f(x), we define the Fourierseries F[h](x) of h(x) to be the Fourier sine series Fs[f](x) off(x), i.e.,we set Fs[f](x) = F[h](x). Explicitly,

Fs[f](x) =

n=1

bn sinnx

L

,(1.4)

where

bn =2

L

L

0

f(x)sinnx

Ldx, n = 1, 2, 3, .

Remark 1. It is sometimes desired that the function f(x) be repre-sented by a Fourier series of period L. In this case, we extend f(x) to

the function

(1.5) k(x) = f(x), 0 x < L,

and require that k(x + L) = k(x) for all x R. Then the Fourierseries F[k](x) of k(x) approximates f(x) inside the original interval.See problem 20 for an example.

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10.4: EVEN AND ODD FUNCTIONS 3

Appendix A. Even and Odd Functions

Recall that a function f(x) is called even on an interval I, where I iseither one of the intervals (L, L), (L, L], [L, L), and [L, L], if

(A.1) f(x) = f(x)

for almost all x I in the sense that (A.1) holds for all x I exceptpossibly at a finite number of points. Geometrically, the graph of f(x)over the interval I is symmetric about the y-axis for the most part inthis case.

Likewise, a function g(x) is called odd on the interval I if

(A.2) g(x) = g(x)

for almost all x I. Geometrically, the graph of g(x) over the intervalI is symmetric about the origin for the most part in this case.

Some useful properties of even and odd functions are as follows.

The product of two odd functions is an even function. The product of two even functions is an even function. The product of an odd function and an even function is an odd

function. Iff(x) is an even function on I, then

L

Lf(x) dx = 2

L

0f(x) dx.

If g(x) is an odd function on I, thenL

Lg(x) dx = 0.

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