1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

514 CHAPTER. 12 • FOURIER. SER.JES AND THE LAPLACE TRANSFORM

Familiar examples of real functions that have period 211" are sin nt and cos nt,
where n is an integer. These examples raise the question of whether any periodic
function can be represented by a sum of terms involving an cos nt and bn sin nt,
where a,. and bn are real constants. As we soon demonstrate, the answer to this
question is often yes.

Definition 12. 1: Piecewise continuous

The function U is piecewise continuous on the closed interval [a,b] if there
exist values to, t1, ... , tn with a = to < ti < · · · < tn = b such that U is continu-
ous in each of the open intervals tk-1 < t < t k (k = 1, 2, ... , n) and has left- and
right-hand limits at the values tk (k = 0, 1, ... , n).

We use the symbols U (a- ) and U (a+) for the left-a nd right-hand limits,

respectively, of a function U (t) as t approaches the point a. The graph of a

piecewise continuous function is illustrated in Figure 12.2, where the function

U (t) is

{

2 ( 3 t - 2 1 ) 2 + 4• l

U(t) = ~-(t- 2 ) 2 ,

1 + t - 3

4 '

~ -(t -5)

3

'

when 1 '.S t < 2;

when 2 < t < 3;

when 3 < t < 4;

when 4 < t ::; 6.

The left-and right-hand limits a t to = 2, t 1 = 3, and t2 = 4 are easily

determined:

7

At t = 2, we have U (2-) =

4

and U(2+)=~·

3

At t = 3, we have U (3-) =

2

and u (3+) = 1.

At t = 4, we have U(4- ) = ~ and U(4+)=1sl·

$

2 3 4 5 6

Figure 12.2 A piecewise continuous function U over the interval [1, 6].