Mathematical Methods for Physicists: A concise introduction - Site Map

THE FOURIER INTEGRAL AND THE DELTA FUNCTION
The Fourier integral and the delta function
The delta function is a very useful tool in physics, but it is not a function in the
usual mathematical sense. The need for this strange `function' arises naturally
from the Fourier integrals. Let us go back to Eqs. (4.30) and (4.31) and substitute
g…!† into f …x†; we then have
f …x† ˆ 1 Z 1 Z 1
d! dx 0 f …x 0 †e i!…xx 0† :
2 1 1
Interchanging the order of integration gives
f …x† ˆ
Z 1
1
dx 0 f …x 0 † 1 Z 1
d!e i!…xx 0† :
2 1
…4:46†
If the above equation holds for any function f …x†, then this tells us something
remarkable about the integral
Z
1 1
d!e i!…xx 0 †
2 1
considered as a function of x 0 . It vanishes everywhere except at x 0 ˆ x, and its
integral with respect to x 0 over any interval including x is unity. That is, we may
think of this function as having an in®nitely high, in®nitely narrow peak at
x ˆ x 0 . Such a strange function is called Dirac's delta function (®rst introduced
by Paul A. M. Dirac):
…x x 0 †ˆ 1 Z 1
d!e i!…xx 0† :
2 1
…4:47†
Equation (4.46) then becomes
f …x† ˆ
Z 1
1
f …x 0 †…x x 0 †dx 0 :
…4:48†
Equation (4.47) is an integral representation of the delta function. We summarize
its properties below:
…x x 0 †ˆ0; if x 0 6ˆ x; …4:49a†
Z b
a
…x x 0 †dx 0 0; if x > b or x < a
ˆ ;
1; if a < x < b
…4:49b†
f …x† ˆ
Z 1
1
f …x 0 †…x x 0 †dx 0 :
…4:49c†
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