Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
It is often convenient to place the origin at the singular point, in which case the
delta function may be written as
…x† ˆ 1
2
Z 1
1
d!e i!x :
…4:50†
To examine the behavior of the function for both small and large x, we use an
alternative representation of this function obtained by integrating as follows:
…x† ˆ 1 Z a
2 lim e i!x 1 e iax e iax
sin ax
d! ˆ lim
ˆ lim
a!1
a!1 2 ix
a!1 x ; …4:51†
a
where a is positive and real. We see immediately that …x† ˆ…x†. To examine
its behavior for small x, we consider the limit as x goes to zero:
sin ax
lim
x!0 x
ˆ a
lim
x!0
sin ax
ax
ˆ a
:
Thus, …0† ˆlim a!1 …a=† !1, or the amplitude becomes in®nite at the singularity.
For large jxj, we see that sin…ax†=x oscillates with period 2=a, and its
amplitude falls o€ as 1=jxj. But in the limit as a goes to in®nity, the period
becomes in®nitesimally narrow so that the function approaches zero everywhere
except for the in®nite spike of in®nitesimal width at the singularity. What is the
integral of Eq. (4.51) over all space?
Z 1
lim
1
a!1
sin ax
x
dx ˆ lim 2
a!1
Z 1
0
sin ax
x dx ˆ 2
2 ˆ 1:
Thus, the delta function may be thought of as a spike function which has unit area
but a non-zero amplitude at the point of singularity, where the amplitude becomes
in®nite. No ordinary mathematical function with these properties exists. How do
we end up with such an improper function? It occurs because the change of order
of integration in Eq. (4.46) is not permissible. In spite of this, the Dirac delta
function is a most convenient function to use symbolically. For in applications the
delta function always occurs under an integral sign. Carrying out this integration,
using the formal properties of the delta function, is really equivalent to inverting
the order of integration once more, thus getting back to a mathematically correct
expression. Thus, using Eq. (4.49) we have
Z 1
1
f …x†…x x 0 †dx ˆ f …x 0 †;
but, on substituting Eq. (4.47) for the delta function, the integral on the left hand
side becomes
Z 1
Z
1 1
f …x† p d!e i!…xx 0 †
dx
2
1
1
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