1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.2 Arbitrary Period and Half-Range Expansions 67

Figure 3 f(x)=x,− 1 <x<1,fperiodic with period 2.

Definition
Afunctiong(x)isevenifg(−x)=g(x);h(x)isoddifh(−x)=−h(x).Note
that a function must be defined on a symmetric interval, say−c<x<c
(wherecmight be∞), in order to qualify as even or odd.

An even function is often said to be symmetric about the vertical axis, and an
odd function is said to be symmetric in the origin. Many familiar functions are
either even or odd. For example, sin(kx),x,x^3 , and any other odd power ofx
are all odd functions defined on the interval−∞<x<∞. Similarly, cos(kx),
|x|,1(=x^0 ),x^2 , and any other even power ofxare even functions over the
same interval. Most functions are neither even nor odd, but any function that
is defined on a symmetric interval can be written as a sum of an even and an
odd function:

f(x)=^1

2

(

f(x)+f(−x)

)

+^1

2

(

f(x)−f(−x)

)

.

It is easy to show that the first term is an even function and the second is odd.
Even and odd functions preserve their symmetries in some algebraic opera-
tions, as summarized here:

even+even=even, odd+odd=odd,

even×even=even, odd×odd=even, odd×even=odd.

We are also concerned with definite integrals of even and odd functions over
symmetric intervals. The symmetry properties lead to important simplifica-
tions in our calculations.

Theorem 1.Let g(x)be an even function defined in a symmetric interval
−a<x<a. Then
∫a

−a

g(x)dx= 2

∫a

0

g(x)dx.