Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Figure 4.17.
Fourier transform of e ikx ; jxj a:
Fourier sine and cosine transforms
If f …x† is an odd function, the Fourier transforms reduce to
r
Z r
2 1
Z
g…!† ˆ f …x 0 † sin !x 0 dx 0 2 1
; f …x† ˆ g…!† sin !xd!:
0
0
…4:33a†
Similarly, if f …x† is an even function, then we have Fourier cosine transformations:
r
Z r
2 1
Z
g…!† ˆ f …x 0 † cos !x 0 dx 0 2 1
; f …x† ˆ g…!† cos !xd!: …4:33b†
0
To demonstrate these results, we ®rst expand the exponential function on the
right hand side of Eq. (4.30)
g…!† ˆp
1
2
ˆ p 1
2
Z 1
1
Z 1
1
f …x 0 †e i!x 0 dx 0
f …x 0 † cos !x 0 dx 0 p
i
2
0
Z 1
1
f …x 0 † sin !x 0 dx 0 :
If f …x† is even, then f …x† cos !x is even and f …x† sin !x is odd. Thus the second
integral on the right hand side of the last equation is zero and we have
g…!† ˆ 1 Z r
1
Z
p f …x 0 † cos !x 0 dx 0 2 1
ˆ f …x 0 † cos !x 0 dx 0 ;
2
1
g…!† is an even function, since g…!† ˆg…!†. Next from Eq. (4.31) we have
f …x† ˆp
1
2
ˆ p 1
2
Z 1
1
Z 1
1
g…!†e i!x d!
g…!† cos !xd! ‡ p
172
i
2
0
Z 1
1
g…!† sin !xd!: