Fourier Series and

Integrals CHAPTER

1

1.1 Periodic Functions and Fourier Series

Afunctionfis said to beperiodic with period p>0 if: (1)f(x)has been de-
fined for allx; and (2)f(x+p)=f(x)for allx. The familiar functions sin(x)
and cos(x)are simple examples of periodic functions with period 2π,andthe
functions sin( 2 πx/p)and cos( 2 πx/p)are periodic with periodp.
A periodic function has many periods, for iff(x)=f(x+p)then also

f(x)=f(x+p)=f(x+ 2 p)=···=f(x+np),

wherenis any integer. Thus sin(x)hasperiods2π, 4 π,..., 2 nπ,....Thepe-
riod of a periodic function is generally taken to be positive, but the periodicity
condition holds for negative as well as positive changes in the argument. That
is to say,f(x−p)=f(x)for allx,sincef(x)=f(x−p+p)=f(x−p).Also,

f(x)=f(x−p)=f(x− 2 p)=···=f(x−np).

The definition of periodic says essentially that functional values repeat
themselves. This implies that the graph of a periodic function can be drawn
for allxby making a template of the graph on any interval of lengthpand
then copying the graph from the template up and down thex-axis (see Fig. 1).
Many of the functions that occur in engineering and physics are periodic
in space or time — for example, acoustic waves — and in order to understand
them better it is often desirable to represent them in terms of the very simple
periodic functions 1, sin(x),cos(x),sin( 2 x),cos( 2 x), and so forth.Allof these

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