Mathematical Methods for Physicists: A concise introduction - Site Map

CALCULATIONS OF FOURIER TRANSFORMS
provided that u and @u=@x ! 0asx !1. In general, we can show that
F
@n u
@x n ˆ…i† n F…u†
if u;@u=@x; ...;@ n1 u=@x n1 !1as x !1.
(c) By de®nition
F
@u Z 1
@u
ˆ
@t 1 @t eix dx ˆ @ Z 1
ue ix dx ˆ @
@t 1
@t F…u†:
Example 4.12
Solve the inhomogeneous di€erential equation
!
d 2
dx 2 ‡ p d dx ‡ q f …x† ˆR…x†; 1x 1;
where p and q are constants.
Solution:
We transform both sides
( )
F
d2 f
dx 2 ‡ p df
dx ‡ qf ˆ‰…i† 2 ‡ p…i†‡qŠFff…x†
g
ˆ FfR…x†
g:
If we denote the Fourier transforms of f …x† and R…x† by g…† and G…†, respectively,
we have
and hence
Fff…x†
g ˆ g…†; FfR…x†
g ˆ G…†;
… 2 ‡ ip ‡ q†g…† ˆG…†; or g…† ˆG…†=… 2 ‡ ip ‡ q†
f …x† ˆp
1
2
ˆ p 1
2
Z 1
1
Z 1
e ix
1
e ix g…†d
G…†
2 ‡ ip ‡ q d:
We will not gain anything if we do not know how to evaluate this complex
integral. This is not a dicult problem in the theory of functions of complex
variables (see Chapter 7).
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