Defined as the limit of a class of Delta Sequences. Sometimes called the Impulse Symbol.
The most commonly used (equivalent) definitions are
![\begin{displaymath}
\delta(x) \equiv \lim_{n\to \infty} {1\over 2\pi} {\sin[(n+{\textstyle{1\over 2}})x]\over \sin({\textstyle{1\over 2}}x)}
\end{displaymath}](d1_424.gif) |
(1) |
(the so-called Dirichlet Kernel) and
where
is the Fourier Transform. Some identities include
![\begin{displaymath}
\delta(x-a) = 0
\end{displaymath}](d1_430.gif) |
(5) |
for
,
![\begin{displaymath}
\int^{a+\epsilon}_{a-\epsilon} \delta (x-a)\,dx = 1,
\end{displaymath}](d1_432.gif) |
(6) |
where
is any Positive number, and
![\begin{displaymath}
\int_{-\infty}^\infty f(x)\delta (x-a)\,dx = f(a)
\end{displaymath}](d1_433.gif) |
(7) |
![\begin{displaymath}
\int_{-\infty}^\infty f(x)\delta'(x-a)dx = -f'(a)
\end{displaymath}](d1_434.gif) |
(8) |
![\begin{displaymath}
(\delta'*f)(a)=\int_{-\infty}^\infty \delta'(a-x)f(x)\,dx=f'(a),
\end{displaymath}](d1_435.gif) |
(9) |
where
denotes Convolution,
![\begin{displaymath}
\int_{-\infty}^\infty \vert\delta'(x)\vert\,dx=\infty
\end{displaymath}](d1_436.gif) |
(10) |
![\begin{displaymath}
x^2\delta'(x)=0
\end{displaymath}](d1_437.gif) |
(11) |
![\begin{displaymath}
\delta'(-x)=-\delta'(x)
\end{displaymath}](d1_438.gif) |
(12) |
![\begin{displaymath}
x\delta'(x)=-\delta(x).
\end{displaymath}](d1_439.gif) |
(13) |
(13) can be established using Integration by Parts as follows:
Additional identities are
![\begin{displaymath}
\delta(ax)={1\over\vert a\vert}\delta(x)
\end{displaymath}](d1_444.gif) |
(15) |
![\begin{displaymath}
\delta(x^2-a^2)={1\over 2\vert a\vert} [\delta(x+a)+\delta(x-a)]
\end{displaymath}](d1_445.gif) |
(16) |
![\begin{displaymath}
\delta[g(x)] = \sum_i {\delta(x-x_i)\over\vert g'(x_i)\vert},
\end{displaymath}](d1_446.gif) |
(17) |
where the
s are the Roots of
. For example, examine
![\begin{displaymath}
\delta(x^2+x-2)=\delta[(x-1)(x+2)].
\end{displaymath}](d1_448.gif) |
(18) |
Then
, so
and
, and we have
![\begin{displaymath}
\delta(x^2+x-2)={\textstyle{1\over 3}}\delta(x-1)+{\textstyle{1\over 3}}\delta(x+2).
\end{displaymath}](d1_452.gif) |
(19) |
A Fourier Series expansion of
gives
![\begin{displaymath}
a_n = {1\over\pi} \int^\pi_{-\pi} \delta(x-a)\cos(nx)\,dx = {1\over\pi}\cos(na)
\end{displaymath}](d1_454.gif) |
(20) |
![\begin{displaymath}
b_n = {1\over\pi} \int^\pi_{-\pi} \delta(x-a)\sin(nx)\,dx = {1\over\pi}\sin(na),
\end{displaymath}](d1_455.gif) |
(21) |
so
The Fourier Transform of the delta function is
![\begin{displaymath}
{\mathcal F}[\delta(x-x_0)] = \int_{-\infty}^\infty e^{-2\pi ikx}\delta(x-x_0)\,dx = e^{-2\pi ikx_0}.
\end{displaymath}](d1_459.gif) |
(23) |
Delta functions can also be defined in 2-D, so that in 2-D Cartesian Coordinates
![\begin{displaymath}
\delta^2(x-x_0,y-y_0)=\delta(x-x_0)\delta(y-y_0),
\end{displaymath}](d1_460.gif) |
(24) |
and in 3-D, so that in 3-D Cartesian Coordinates
![\begin{displaymath}
\delta^3(x-x_0,y-y_0,z-z_0)=\delta (x-x_0)\delta(y-y_0)\delta(z-z_0),
\end{displaymath}](d1_461.gif) |
(25) |
in Cylindrical Coordinates
![\begin{displaymath}
\delta^3(r,\theta,z) = {\delta(r)\delta(z)\over\pi r},
\end{displaymath}](d1_462.gif) |
(26) |
and in Spherical Coordinates,
![\begin{displaymath}
\delta^3(r,\theta,\phi)={\delta(r)\over 2\pi r^2}.
\end{displaymath}](d1_463.gif) |
(27) |
A series expansion in Cylindrical Coordinates gives
|
|
|
|
|
(28) |
The delta function also obeys the so-called Sifting Property
![\begin{displaymath}
\int f({\bf y})\delta({\bf x}-{\bf y})\,d{\bf y}= f({\bf x}).
\end{displaymath}](d1_466.gif) |
(29) |
See also Delta Sequence, Doublet Function, Fourier Transform--Delta Function
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 481-485, 1985.
Spanier, J. and Oldham, K. B. ``The Dirac Delta Function
.''
Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79-82, 1987.
© 1996-9 Eric W. Weisstein
1999-05-24