Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER INTEGRALS AND FOURIER TRANSFORMS
which has period L > 2. Fig. 4.11(a) shows the function when L ˆ 4. If L is
increased to 8, the function looks like the one shown in Fig. 4.11(b). As
L !1we obtain a non-periodic function f …x†, as shown in Fig. 4.11(c):
f …x† ˆ 1 1 < x < 1 :
0 otherwise
Example 4.6
Consider the periodic function g L …x† (Fig. 4.12(a)):
g L …x† ˆe jxj when L=2 < x < L=2:
As L !1 we obtain a non-periodic function g…x†: g…x† ˆlim L!1 g L …x† (Fig.
4.12(b)).
By investigating the limit that is approached by a Fourier series as the period of
the given function becomes in®nite, a suitable representation for non-periodic
functions can perhaps be obtained. To this end, let us write the Fourier series
representing a periodic function f …x† in complex form:
where ! denotes n=L
f …x† ˆ X1
nˆ1
c n e i!x ;
c n ˆ 1 Z L
f …x†e i!x dx
2L L
…4:25†
…4:26†
! ˆ n ; n positive or negative: …4:27†
L
The transition L !1is a little tricky since c n apparently approaches zero, but
these coecients should not approach zero. We can ask for help from Eq. (4.27),
from which we have
! ˆ…=L†n;
Figure 4.12. Sawtooth wave functions: …a† L=2 < x < L=2; …b† L !1.
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