# §1.17(i) Delta Sequences

In applications in physics and engineering, the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\mathop{\delta\/}\nolimits\!\left(x\right)$. This is an operator with the properties:

 1.17.1 $\mathop{\delta\/}\nolimits\!\left(x\right)=0,$ $x\in\Real$, $x\neq 0$,

and

 1.17.2 $\int_{-\infty}^{\infty}\mathop{\delta\/}\nolimits\!\left(x-a\right)\phi(x)dx=% \phi(a),$ $a\in\Real$,

subject to certain conditions on the function $\phi(x)$. From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that

 1.17.3 $\lim_{n\to\infty}\int_{-\infty}^{\infty}\mathop{\delta_{n}\/}\nolimits\!\left(% x-a\right)\phi(x)dx=\phi(a),$

for a suitably chosen sequence of functions $\mathop{\delta_{n}\/}\nolimits\!\left(x\right)$, $n=1,2,\dots$. Such a sequence is called a delta sequence and we write, symbolically,

 1.17.4 $\lim_{n\to\infty}\mathop{\delta_{n}\/}\nolimits\!\left(x\right)=\mathop{\delta% \/}\nolimits\!\left(x\right),$ $x\in\Real$.

An example of a delta sequence is provided by

 1.17.5 $\mathop{\delta_{n}\/}\nolimits\!\left(x-a\right)=\sqrt{\frac{n}{\pi}}e^{-n(x-a% )^{2}}.$

In this case

 1.17.6 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}% \phi(x)dx=\phi(a),$

for all functions $\phi(x)$ that are continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)dx$ converges absolutely for all sufficiently large values of $n$. The last condition is satisfied, for example, when $\phi(x)=\mathop{O\/}\nolimits\!\left(e^{\alpha x^{2}}\right)$ as $x\to\pm\infty$, where $\alpha$ is a real constant.

More generally, assume $\phi(x)$ is piecewise continuous (§1.4(ii)) when $x\in[-c,c]$ for any finite positive real value of $c$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)dx$ converges absolutely for all sufficiently large values of $n$. Then

 1.17.7 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}% \phi(x)dx=\tfrac{1}{2}\phi(a-)+\tfrac{1}{2}\phi(a+).$

# §1.17(ii) Integral Representations

Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)):

 1.17.8 $\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iat}\left(\int_{-\infty}^{\infty}\phi% (x)e^{itx}dx\right)dt=\phi(a)$

yields

 1.17.9 $\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}% dt\right)\phi(x)dx=\phi(a).$

The inner integral does not converge. However, for $n=1,2,\dots$,

 1.17.10 $\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-t^{2}/(4n)}e^{i(x-a)t}dt=\sqrt{\frac{% n}{\pi}}e^{-n(x-a)^{2}}.$

Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with

 1.17.11 $\mathop{\delta_{n}\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^% {\infty}e^{-t^{2}/(4n)}e^{i(x-a)t}dt,$

provided that $\phi(x)$ is continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation

 1.17.12 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{% \infty}e^{i(x-a)t}dt.$

Other similar integral representations of the Dirac delta that appear in the physics literature include the following:

# Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))

 1.17.13 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=x\int_{0}^{\infty}t\mathop{J_{\nu% }\/}\nolimits\!\left(xt\right)\mathop{J_{\nu}\/}\nolimits\!\left(at\right)dt,$ $\realpart{\nu}>-1$, $x>0$, $a>0$,
 1.17.14 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{2xa}{\pi}\int_{0}^{\infty}t% ^{2}\mathop{\mathsf{j}_{\ell}\/}\nolimits\!\left(xt\right)\mathop{\mathsf{j}_{% \ell}\/}\nolimits\!\left(at\right)dt,$ $x>0$, $a>0$.

See Arfken and Weber (2005, Eq. (11.59)) and Konopinski (1981, p. 242). For a generalization of (1.17.14) see Maximon (1991).

# Coulomb Functions (§33.14(iv))

 1.17.15 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\int_{0}^{\infty}\mathop{s\/}% \nolimits\!\left(x,\ell;r\right)\mathop{s\/}\nolimits\!\left(a,\ell;r\right)dr,$ $a>0$, $x>0$.

See Seaton (2002a).

# Airy Functions (§9.2)

 1.17.16 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\int_{-\infty}^{\infty}\mathop{% \mathrm{Ai}\/}\nolimits\!\left(t-x\right)\mathop{\mathrm{Ai}\/}\nolimits\!% \left(t-a\right)dt.$

See Vallée and Soares (2010, §3.5.3).

# §1.17(iii) Series Representations

Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)):

 1.17.17 $\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{-ika}\left(\int_{-\pi}^{\pi}\phi(x)e% ^{ikx}dx\right)=\phi(a),$

yields

 1.17.18 $\int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{ik(x-a% )}\right)dx=\phi(a).$

The sum $\sum_{k=-\infty}^{\infty}e^{ik(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that

 1.17.19 $\lim_{n\to\infty}\int_{-\pi}^{\pi}\mathop{\delta_{n}\/}\nolimits\!\left(x-a% \right)\phi(x)dx=\phi(a),$

where

 1.17.20 $\mathop{\delta_{n}\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-n}^{n}% e^{ik(x-a)}\left(=\frac{\mathop{\sin\/}\nolimits\!\left((n+\frac{1}{2})(x-a)% \right)}{2\pi\mathop{\sin\/}\nolimits\!\left(\frac{1}{2}(x-a)\right)}\right),$

provided that $\phi(x)$ is continuous and of period $2\pi$; see §1.8(ii).

By analogy with §1.17(ii) we have the formal series representation

 1.17.21 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-\infty}^{% \infty}e^{ik(x-a)}.$

Other similar series representations of the Dirac delta that appear in the physics literature include the following:

# Legendre Polynomials (§§14.7(i) and 18.3)

 1.17.22 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\sum_{k=0}^{\infty}(k+\tfrac{1}{2% })\mathop{P_{k}\/}\nolimits\!\left(x\right)\mathop{P_{k}\/}\nolimits\!\left(a% \right).$

# Laguerre Polynomials (§18.3)

 1.17.23 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=e^{-(x+a)/2}\sum_{k=0}^{\infty}% \mathop{L_{k}\/}\nolimits\!\left(x\right)\mathop{L_{k}\/}\nolimits\!\left(a% \right).$

# Hermite Polynomials (§18.3)

 1.17.24 $\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{e^{-(x^{2}+a^{2})/2}}{\sqrt% {\pi}}\sum_{k=0}^{\infty}\frac{\mathop{H_{k}\/}\nolimits\!\left(x\right)% \mathop{H_{k}\/}\nolimits\!\left(a\right)}{2^{k}k!}.$

# Spherical Harmonics (§14.30)

 1.17.25 $\mathop{\delta\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta_{1}-\mathop{% \cos\/}\nolimits\theta_{2}\right)\mathop{\delta\/}\nolimits\!\left(\phi_{1}-% \phi_{2}\right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\mathop{Y_{{\ell},{% m}}\/}\nolimits\!\left(\theta_{1},\phi_{1}\right){\mathop{Y_{{\ell},{m}}\/}% \nolimits^{\ast}}\!\left(\theta_{2},\phi_{2}\right).$

(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). For (1.17.25) see Arfken and Weber (2005, p. 792).

# §1.17(iv) Mathematical Definitions

The references given in §§1.17(ii)1.17(iii) are from the physics literature. For mathematical interpretations of (1.17.13), (1.17.15), (1.17.16) and (1.17.22)–(1.17.25) that resemble those given in §§1.17(ii) and 1.17(iii) for (1.17.12) and (1.17.21), see Li and Wong (2008). For (1.17.14) combine (1.17.13) and (10.47.3).