Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Figure 4.13. Gaussian probability function: …a† large ; …b† small .
It is easy to see that g…!† is also a Gaussian probability function with a peak at the
origin, monotonically decreasing as ! !1. Furthermore, for large , f …x† is
sharply peaked but g…!† is ¯attened, and vice versa as shown in Fig. 4.13. It is
interesting to note that this is a general feature of Fourier transforms. We shall see
later that in quantum mechanical applications it is related to the Heisenberg
uncertainty principle.
The original function f …x† can be retrieved from Eq. (4.31) which takes the
form
Z
1 1
p g…!†e i!x d! ˆ 1 Z
N 1
p p e !2 =4 e i!x d!
2 1
2 2 1
ˆ 1 Z
N 1
p p e 0 ! 2 e i!x 0 d!
2 2
in which we have set 0 ˆ 1=4, and x 0 ˆx. The last integral can be evaluated
by the same technique, and we ®nally ®nd
1 Z 1
p g…!†e i!x d! ˆ 1 Z
N 1
p p e 0 ! 2 e i!x 0 d!
2 1
2 2 1
ˆ N p
p 2 e
x 2
2
1
ˆ Ne x2 ˆ f …x†:
Example 4.8
Given the box function which can represent a single pulse
f …x† ˆ 1 jxja
0 jxj > a
®nd the Fourier transform of f …x†, g…!†; then graph f …x† and g…!† for a ˆ 3.
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