info prev up next book cdrom email home

Delta Function

Defined as the limit of a class of Delta Sequences. Sometimes called the Impulse Symbol. The most commonly used (equivalent) definitions are

\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} (1)

(the so-called Dirichlet Kernel) and
$\displaystyle \delta(x)$ $\textstyle \equiv$ $\displaystyle \lim_{n\to \infty} {\sin(nx)\over\pi x}$ (2)
  $\textstyle =$ $\displaystyle {1\over 2\pi} \int_{-\infty}^\infty e^{-ikx}\,dk$ (3)
  $\textstyle =$ $\displaystyle {\mathcal F}[1],$ (4)

where ${\mathcal F}$ is the Fourier Transform. Some identities include
\delta(x-a) = 0
\end{displaymath} (5)

for $x \not= a$,
\int^{a+\epsilon}_{a-\epsilon} \delta (x-a)\,dx = 1,
\end{displaymath} (6)

where $\epsilon$ is any Positive number, and
\int_{-\infty}^\infty f(x)\delta (x-a)\,dx = f(a)
\end{displaymath} (7)

\int_{-\infty}^\infty f(x)\delta'(x-a)dx = -f'(a)
\end{displaymath} (8)

(\delta'*f)(a)=\int_{-\infty}^\infty \delta'(a-x)f(x)\,dx=f'(a),
\end{displaymath} (9)

where $*$ denotes Convolution,
\int_{-\infty}^\infty \vert\delta'(x)\vert\,dx=\infty
\end{displaymath} (10)

\end{displaymath} (11)

\end{displaymath} (12)

\end{displaymath} (13)

(13) can be established using Integration by Parts as follows:
$\displaystyle \int f(x)x\delta'(x)\,dx$ $\textstyle =$ $\displaystyle -\int \delta(x){d\over dx} [xf(x)]\,dx$  
  $\textstyle =$ $\displaystyle -\int \delta[f(x)+xf'(x)]\,dx$  
  $\textstyle =$ $\displaystyle -\int f(x)\delta(x)\, dx.$ (14)

Additional identities are
\delta(ax)={1\over\vert a\vert}\delta(x)
\end{displaymath} (15)

\delta(x^2-a^2)={1\over 2\vert a\vert} [\delta(x+a)+\delta(x-a)]
\end{displaymath} (16)

\delta[g(x)] = \sum_i {\delta(x-x_i)\over\vert g'(x_i)\vert},
\end{displaymath} (17)

where the $x_i$s are the Roots of $g$. For example, examine
\end{displaymath} (18)

Then $g'(x)=2x+1$, so $g'(x_1)=g'(1)=3$ and $g'(x_2)=g'(-2)=-3$, and we have
\delta(x^2+x-2)={\textstyle{1\over 3}}\delta(x-1)+{\textstyle{1\over 3}}\delta(x+2).
\end{displaymath} (19)

A Fourier Series expansion of $\delta(x-a)$ gives

a_n = {1\over\pi} \int^\pi_{-\pi} \delta(x-a)\cos(nx)\,dx = {1\over\pi}\cos(na)
\end{displaymath} (20)

b_n = {1\over\pi} \int^\pi_{-\pi} \delta(x-a)\sin(nx)\,dx = {1\over\pi}\sin(na),
\end{displaymath} (21)

$\displaystyle \delta(x-a)$ $\textstyle =$ $\displaystyle {1\over 2\pi} + {1\over \pi} \sum_{n=1}^\infty [\cos(na)\cos(nx)+\sin(na)\sin(nx)]$  
  $\textstyle =$ $\displaystyle {1\over 2\pi} + {1\over \pi} \sum_{n=1}^\infty \cos[n(x-a)].$ (22)

The Fourier Transform of the delta function is
{\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} (23)

Delta functions can also be defined in 2-D, so that in 2-D Cartesian Coordinates
\end{displaymath} (24)

and in 3-D, so that in 3-D Cartesian Coordinates
\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} (25)

in Cylindrical Coordinates
\delta^3(r,\theta,z) = {\delta(r)\delta(z)\over\pi r},
\end{displaymath} (26)

and in Spherical Coordinates,
\delta^3(r,\theta,\phi)={\delta(r)\over 2\pi r^2}.
\end{displaymath} (27)

A series expansion in Cylindrical Coordinates gives
$\delta^3({\bf r}_1-{\bf r}_2) = {1\over r_1}\delta(r_1-r_2)\delta(\phi_1-\phi_2)\delta(z_1-z_2)$
$= {1\over r_1}\delta(r_1-r_2){1\over 2\pi} \sum_{m=-\infty}^\infty \!\!e^{im(\phi_1-\phi_2)}{1\over 2\pi}\int_{-\infty}^\infty e^{ik(z_1-z_2)}\,dk.$


The delta function also obeys the so-called Sifting Property

\int f({\bf y})\delta({\bf x}-{\bf y})\,d{\bf y}= f({\bf x}).
\end{displaymath} (29)

See also Delta Sequence, Doublet Function, Fourier Transform--Delta Function


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 $\delta(x-a)$.'' Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79-82, 1987.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein