Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
and the `adjacent' values of ! are obtained by setting n ˆ 1, which corresponds
to
…L=†! ˆ 1:
Then we can multiply each term of the Fourier series by …L=†! and obtain
f …x† ˆ X1 L
c n e i!x !;
where
nˆ1
L
c n ˆ 1 Z L
f …x†e i!x dx:
2 L
The troublesome factor 1=L has disappeared. Switching completely to the !
notation and writing …L=†c n ˆ c L …!†, we obtain
and
c L …!† ˆ 1
2
Z L
L
f …x†e i!x dx
f …x† ˆ
X1
L!=ˆ1
c L …!†e i!x !:
In the limit as L !1, the !s are distributed continuously instead of discretely,
! ! d! and this sum is exactly the de®nition of an integral. Thus the last
equations become
c…!† ˆ lim c L …!† ˆ 1 Z 1
f …x†e i!x dx
…4:28†
L!1 2 1
and
f …x† ˆ
Z 1
1
c…!†e i!x d!:
…4:29†
This set of formulas is known as the Fourier transformation, in somewhat di€erent
form. It is easy to put them in a symmetrical form by de®ning
p
g…!† ˆ
2 c…!†;
then Eqs. (4.28) and (4.29) take the symmetrical form
g…!† ˆp
1
2
f …x† ˆp
1
2
Z 1
1
Z 1
1
f …x 0 †e i!x 0 dx 0 ;
g…!†e i!x d!:
…4:30†
…4:31†
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