Delta Function

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

$$\delta(x) \equiv \lim_{n\to \infty} {1\over 2\pi} {\sin[(n+{\textstyle{1\over 2}})x]\over \sin({\textstyle{1\over 2}}x)}$$ (1)

(the so-called Dirichlet Kernel) and

$$\delta(x) \equiv \lim_{n\to \infty} {\sin(nx)\over\pi x}$$ (2)

$$= {1\over 2\pi} \int_{-\infty}^\infty e^{-ikx}\,dk$$ (3)

$$= {\mathcal F}[1],$$ (4)

where ${\mathcal F}$ is the Fourier Transform. Some identities include

$$\delta(x-a) = 0$$ (5)

for $x \not= a$,

$$\int^{a+\epsilon}_{a-\epsilon} \delta (x-a)\,dx = 1,$$ (6)

where $\epsilon$ is any Positive number, and

$$\int_{-\infty}^\infty f(x)\delta (x-a)\,dx = f(a)$$ (7)

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

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

where $*$ denotes Convolution,

$$\int_{-\infty}^\infty \vert\delta'(x)\vert\,dx=\infty$$ (10)

$$x^2\delta'(x)=0$$ (11)

$$\delta'(-x)=-\delta'(x)$$ (12)

$$x\delta'(x)=-\delta(x).$$ (13)

(13) can be established using Integration by Parts as follows:

$$\int f(x)x\delta'(x)\,dx = -\int \delta(x){d\over dx} [xf(x)]\,dx$$

$$= -\int \delta[f(x)+xf'(x)]\,dx$$

$$= -\int f(x)\delta(x)\, dx.$$ (14)

Additional identities are

$$\delta(ax)={1\over\vert a\vert}\delta(x)$$ (15)

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

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

where the $x_i$s are the Roots of $g$. For example, examine

$$\delta(x^2+x-2)=\delta[(x-1)(x+2)].$$ (18)

Then $g'(x)=2x+1$, so $g'(x_1)=g'(1)=3$ and $g'(x_2)=g'(-2)=-3$, and we have

$$\delta(x^2+x-2)={\textstyle{1\over 3}}\delta(x-1)+{\textstyle{1\over 3}}\delta(x+2).$$ (19)

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

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

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

so

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

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

The Fourier Transform of the delta function is

$${\mathcal F}[\delta(x-x_0)] = \int_{-\infty}^\infty e^{-2\pi ikx}\delta(x-x_0)\,dx = e^{-2\pi ikx_0}.$$ (23)

Delta functions can also be defined in 2-D, so that in 2-D Cartesian Coordinates

$$\delta^2(x-x_0,y-y_0)=\delta(x-x_0)\delta(y-y_0),$$ (24)

and in 3-D, so that in 3-D Cartesian Coordinates

$$\delta^3(x-x_0,y-y_0,z-z_0)=\delta (x-x_0)\delta(y-y_0)\delta(z-z_0),$$ (25)

in Cylindrical Coordinates

$$\delta^3(r,\theta,z) = {\delta(r)\delta(z)\over\pi r},$$ (26)

and in Spherical Coordinates,

$$\delta^3(r,\theta,\phi)={\delta(r)\over 2\pi r^2}.$$ (27)

A series expansion in Cylindrical Coordinates gives

$\delta^3({\bf r}_1-{\bf r}_2) = {1\over r_1}\delta(r_1-r_2)\delta(\phi_1-\phi_2)\delta(z_1-z_2)$
$= {1\over r_1}\delta(r_1-r_2){1\over 2\pi} \sum_{m=-\infty}^\infty \!\!e^{im(\phi_1-\phi_2)}{1\over 2\pi}\int_{-\infty}^\infty e^{ik(z_1-z_2)}\,dk.$
	(28)

The delta function also obeys the so-called Sifting Property

$$\int f({\bf y})\delta({\bf x}-{\bf y})\,d{\bf y}= f({\bf x}).$$ (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 $\delta(x-a)$.'' Ch. 10 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 79-82, 1987.