§1.17 Integral and Series Representations of the Dirac Delta
Contents
- §1.17(i) Delta Sequences
- §1.17(ii) Integral Representations
- §1.17(iii) Series Representations
- §1.17(iv) Mathematical Definitions
§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)
. This is an operator with the properties:
![\mathop{\delta\/}\nolimits\!\left(x\right)=0,](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E1.png)
![x\in\Real](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/m46.png)
![x\neq 0](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/m47.png)
and
![\int _{{-\infty}}^{{\infty}}\mathop{\delta\/}\nolimits\!\left(x-a\right)\phi(x)dx=\phi(a),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E2.png)
![a\in\Real](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/m33.png)
subject to certain conditions on the function . From the mathematical
standpoint the left-hand side of (1.17.2) can be interpreted as
a generalized integral in the sense that
![\lim _{{n\to\infty}}\int _{{-\infty}}^{{\infty}}\mathop{\delta _{{n}}\/}\nolimits\!\left(x-a\right)\phi(x)dx=\phi(a),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E3.png)
for a suitably chosen sequence of functions ,
.
Such a sequence is called a delta sequence and we write, symbolically,
![\lim _{{n\to\infty}}\mathop{\delta _{{n}}\/}\nolimits\!\left(x\right)=\mathop{\delta\/}\nolimits\!\left(x\right),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E4.png)
![x\in\Real](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/m46.png)
An example of a delta sequence is provided by
![\mathop{\delta _{{n}}\/}\nolimits\!\left(x-a\right)=\sqrt{\frac{n}{\pi}}e^{{-n(x-a)^{2}}}.](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E5.png)
In this case
![\lim _{{n\to\infty}}\sqrt{\frac{n}{\pi}}\int _{{-\infty}}^{{\infty}}e^{{-n(x-a)^{2}}}\phi(x)dx=\phi(a),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E6.png)
for all functions that are continuous when
,
and for each
,
converges absolutely for all sufficiently large values of
. The last
condition is satisfied, for example, when
as
, where
is a real constant.
More generally, assume is piecewise continuous (§1.4(ii))
when
for any finite positive real value of
, and for each
,
converges absolutely
for all sufficiently large values of
. Then
![\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+).](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E7.png)
§1.17(ii) Integral Representations
Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)):
![\frac{1}{2\pi}\int _{{-\infty}}^{{\infty}}e^{{-iat}}\left(\int _{{-\infty}}^{{\infty}}\phi(x)e^{{itx}}dx\right)dt=\phi(a)](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E8.png)
yields
![\int _{{-\infty}}^{{\infty}}\left(\frac{1}{2\pi}\int _{{-\infty}}^{{\infty}}e^{{i(x-a)t}}dt\right)\phi(x)dx=\phi(a).](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E9.png)
The inner integral does not converge. However, for ,
![\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}}}.](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E10.png)
Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with
![\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,](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E11.png)
provided that is continuous when
, and for
each
,
converges
absolutely for all sufficiently large values of
(as in the case of
(1.17.6)). Then comparison of (1.17.2) and
(1.17.9) yields the formal integral representation
![\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\int _{{-\infty}}^{{\infty}}e^{{i(x-a)t}}dt.](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E12.png)
Other similar integral representations of the Dirac delta that appear in the physics literature include the following:
§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)):
![\frac{1}{2\pi}\sum _{{k=-\infty}}^{{\infty}}e^{{-ika}}\left(\int _{{-\pi}}^{{\pi}}\phi(x)e^{{ikx}}dx\right)=\phi(a),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E17.png)
yields
![\int _{{-\pi}}^{{\pi}}\phi(x)\left(\frac{1}{2\pi}\sum _{{k=-\infty}}^{{\infty}}e^{{ik(x-a)}}\right)dx=\phi(a).](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E18.png)
The sum does not converge, but
(1.17.18) can be interpreted as a generalized integral in the
sense that
![\lim _{{n\to\infty}}\int _{{-\pi}}^{{\pi}}\mathop{\delta _{{n}}\/}\nolimits\!\left(x-a\right)\phi(x)dx=\phi(a),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E19.png)
where
![\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),](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E20.png)
provided that is continuous and of period
; see
§1.8(ii).
By analogy with §1.17(ii) we have the formal series representation
![\mathop{\delta\/}\nolimits\!\left(x-a\right)=\frac{1}{2\pi}\sum _{{k=-\infty}}^{{\infty}}e^{{ik(x-a)}}.](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E21.png)
Other similar series representations of the Dirac delta that appear in the physics literature include the following:
¶ Laguerre Polynomials (§18.3)
![\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).](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E23.png)
¶ Hermite Polynomials (§18.3)
![\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!}.](https://webarchive.library.unt.edu/web/20130221131624im_/http://dlmf.nist.gov/1/17/E24.png)
§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).