Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Figure 4.15.
The Gibb's phenomenon.
Just as in Fourier series expansion, we also expect to observe Gibb's
phenomenon in the case of Fourier integrals. Approximations to the Fourier
integral are obtained by replacing 1 by :
Z
sin ! cos !x
d!;
!
0
where we have set a ˆ 1. Fig. 4.15 shows oscillations near the points of discontinuity
of f …x†. We might expect these oscillations to disappear as !1, but they
are just shifted closer to the points x ˆ1.
Example 4.9
Consider now a harmonic wave of frequency ! 0 , e i! 0t , which is chopped to a lifetime
of 2T seconds (Fig. 4.16(a)):
(
f …t† ˆ
ei! 0t
T t T
:
0 jtj > 0
The chopping process will introduce many new frequencies in varying amounts,
given by the Fourier transform. Then we have, according to Eq. (4.30),
Z T
Z T
g…!† ˆ…2† 1=2 e i! 0t e i!t dt ˆ…2† 1=2 e i…! 0!†t dt
T
ˆ…2† 1=2 e i…! 0!†t
i…! 0 !†
T
T
T
ˆ…2=† 1=2 T sin…! 0 !†T
…! 0 !†T :
This function is plotted schematically in Fig. 4.16(b). (Note that
lim x!0 …sin x=x† ˆ1.) The most striking aspect of this graph is that, although
the principal contribution comes from the frequencies in the neighborhood of
! 0 , an in®nite number of frequencies are presented. Nature provides an example
of this kind of chopping in the emission of photons during electronic and nuclear
transitions in atoms. The light emitted from an atom consists of regular vibrations
that last for a ®nite time of the order of 10 9 s or longer. When light is examined
by a spectroscope (which measures the wavelengths and, hence, the frequencies)
we ®nd that there is an irreducible minimum frequency spread for each spectrum
line. This is known as the natural line width of the radiation.
The relative percentage of frequencies, other than the basic one, present
depends on the shape of the pulse, and the spread of frequencies depends on
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