15.6 Fourier transforms 439
EXAMPLE 15.10Find the Fourier transform of the exponential function, Figure
15.14(c),
f(x) 1 = 1 e
−axfor x 1 > 1 0 and a 1 > 10
By equation (15.82),
The presence of a complex exponent does not alter the normal rule for the integration
of the exponential. Thus
becausee
−(a+iy)x1 = 1 e
−axe
−iyx1 → 1 0asx 1 → 1 ∞. The Fourier transform is complex and to
find the real part we write
Then
(15.83)
This function is called a Lorentzian, and gives the shapes of spectral lines in Fourier
transform spectroscopy.
0 Exercises 15–17
In applications in the physical sciences, the variables xand yare usually a pair of
conjugate variables. The most important of these are the coordinate–momentum
pair, such as the xcoordinate and its conjugate momentum, the linear momentum
p
x, and the time–frequency (or time–energy) pair. A Fourier transformation is then
a transformation of the description of a physical system from one in the space
(or domain) of one of the variables of a conjugate pair to that of the other. The
momentum to coordinate Fourier transformation is an essential tool in the analysis
Regy
a
ay
()=
1
2
22π
gy
aiy
aiyaiy
aiy
ay
()
()()
=
−
+−
=
−
1
2
1
2
22ππ
gy
e
aiy aiy
aiyx()
()=
−
=
−+1
2
1
2
1
0ππ
∞
=
+−+1
2
0π
Z
∞edx
()aiyxgy fxe dx e e dx
ixy ax ixy()==()
−+−−−1
2
1
2
0ππ
ZZ
∞∞∞