Editing Dirac delta function (section)

==History==
In physics, the Dirac delta function was popularized by [[Paul Dirac]] in this book ''[[The Principles of Quantum Mechanics]]'' published in 1930.{{sfn|Dirac|1930|loc=§22 The ''δ'' function}}  However, [[Oliver Heaviside]], 35 years before Dirac, described an impulsive function called the [[Heaviside step]] function for purposes and with properties analogous to Dirac's work. Even earlier several mathematicians and physicists used limits of sharply peaked functions in derivations.<ref name=JacksonHistory>{{Cite journal |last=Jackson |first=J. D. |date=2008-08-01 |title=Examples of the zeroth theorem of the history of science |url=https://pubs.aip.org/aapt/ajp/article-abstract/76/8/704/1057888/Examples-of-the-zeroth-theorem-of-the-history-of?redirectedFrom=fulltext |journal=American Journal of Physics |volume=76 |issue=8 |pages=704–719 |doi=10.1119/1.2904468 |issn=0002-9505|arxiv=0708.4249 |bibcode=2008AmJPh..76..704J }}</ref>
An [[infinitesimal]] formula for an infinitely tall, unit impulse delta function (infinitesimal version of [[Cauchy distribution]]) explicitly appears in an 1827 text of [[Augustin-Louis Cauchy]].{{sfn|Laugwitz|1989|p=230}} [[Siméon Denis Poisson]] considered the issue in connection with the study of wave propagation as did [[Gustav Kirchhoff]] somewhat later. Kirchhoff and [[Hermann von Helmholtz]] also introduced the unit impulse as a limit of [[Gaussian distribution|Gaussians]], which also corresponded to [[Lord Kelvin]]'s notion of a point heat source.<ref>A more complete historical account can be found in {{harvnb|van der Pol|Bremmer|1987|loc=§V.4}}.</ref> The Dirac delta function as such was introduced by [[Paul Dirac]] in his 1927 paper ''The Physical Interpretation of the Quantum Dynamics.''<ref>{{Cite journal |date=January 1927 |title=The physical interpretation of the quantum dynamics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=113 |issue=765 |pages=621–641 |doi=10.1098/rspa.1927.0012 |bibcode=1927RSPSA.113..621D |issn=0950-1207|last1=Dirac |first1=P. A. M. |s2cid=122855515 |doi-access=free }}</ref>  He called it the "delta function" since he used it as a [[Continuum (set theory)|continuum]] analogue of the discrete [[Kronecker delta]].

Mathematicians refer to the same concept as a [[Distribution (mathematics)|distribution]] rather than a function.<ref>{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1st |series=In a Nutshell Series |location=Princeton}}</ref>{{rp|33}}
[[Joseph Fourier]] presented what is now called the [[Fourier integral theorem]] in his treatise ''Théorie analytique de la chaleur'' in the form:<ref name=Fourier>{{cite book |title=The Analytical Theory of Heat |first=JB |last=Fourier |author-link=Joseph Fourier |year=1822 |page=[{{google books |plainurl=y |id=-N8EAAAAYAAJ|page=408}}] |edition= English translation by Alexander Freeman, 1878 |publisher=The University Press}}, cf. {{google books |plainurl=y |id=-N8EAAAAYAAJ|page=449
}} and pp. 546–551. [{{google books |plainurl=y |id=TDQJAAAAIAAJ|page=525}} Original French text].</ref>

<math display="block">f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha \, f(\alpha) \ \int_{-\infty}^\infty dp\ \cos  (px-p\alpha)\ , </math>

which is tantamount to the introduction of the {{mvar|δ}}-function in the form:<ref name= Kawai>{{cite book |title=Microlocal Analysis and Complex Fourier Analysis |editor1=[[Takahiro Kawai]] |editor2=Keiko Fujita |first=Hikosaburo |last=Komatsu |chapter=Fourier's hyperfunctions and Heaviside's pseudodifferential operators |isbn=978-981-238-161-3 |year=2002 |publisher=World Scientific |chapter-url={{google books |plainurl=y |id=8GwKzEemrIcC}}|page=[{{google books |plainurl=y |id=8GwKzEemrIcC|page=200}}]}}</ref>

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos  (px-p\alpha) \ . </math>

Later, [[Augustin Cauchy]] expressed the theorem using exponentials:<ref name= Myint-U>{{cite book |first1=Tyn |last1=Myint-U. |first2=Lokenath |last2=Debnath|author2-link=Lokenath Debnath |title=Linear Partial Differential Equations for Scientists And Engineers|url={{google books |plainurl=y |id=Zbz5_UvERIIC}} |isbn=978-0-8176-4393-5 |edition=4th  |year=2007  |page=[{{google books |plainurl=y |id=Zbz5_UvERIIC|page=4}}] |publisher=Springer}}</ref><ref name=Debnath>{{cite book |title=Integral Transforms And Their Applications |first1=Lokenath |last1=Debnath |first2=Dambaru |last2=Bhatta |isbn=978-1-58488-575-7 |url={{google books |plainurl=y |id=WbZcqdvCEfwC}}|year=2007 |edition=2nd |publisher=[[CRC Press]] |page=[{{google books |plainurl=y |id=WbZcqdvCEfwC|page=2}}]}}</ref>

<math display="block">f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp. </math>

Cauchy pointed out that in some circumstances the ''order'' of integration is significant in this result (contrast [[Fubini's theorem]]).<ref name=Grattan-Guinness>{{cite book |title=Convolutions in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2 |url={{google books |plainurl=y |id=_GgioErrbW8C}}|isbn=978-3-7643-2238-0 |year=2009 |publisher=Birkhäuser |first=Ivor |last=Grattan-Guinness|author-link=Ivor Grattan-Guinness|page=[{{google books |plainurl=y |id=_GgioErrbW8C|page=653}} 653]}}</ref><ref name=Cauchy>

See, for example, {{Cite book|last=Cauchy|first=Augustin-Louis (1789-1857) Auteur du texte|url=https://gallica.bnf.fr/ark:/12148/bpt6k90181x|title=Oeuvres complètes d'Augustin Cauchy. Série 1, tome 1 / publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique...|date=1882–1974|language=EN|chapter-url=http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387 |chapter=Des intégrales doubles qui se présentent sous une forme indéterminèe}}</ref>

As justified using the [[Distribution (mathematics)|theory of distributions]], the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the ''δ''-function as

<math display="block">\begin{align}
f(x)&=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp \\[4pt]
&=\frac{1}{2\pi} \int_{-\infty}^\infty \left(\int_{-\infty}^\infty e^{ipx} e^{-ip\alpha } \,dp \right)f(\alpha)\,d \alpha =\int_{-\infty}^\infty \delta (x-\alpha) f(\alpha) \,d \alpha,
\end{align}</math>

where the ''δ''-function is expressed as

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\,dp \ . </math>

A rigorous interpretation of the exponential form and the various limitations upon the function ''f'' necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:<ref name="Mitrović">{{cite book |title=Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces |first1=Dragiša |last1=Mitrović |first2=Darko |last2=Žubrinić |url={{google books |plainurl=y |id=Od5BxTEN0VsC}}|page=[{{google books |plainurl=y |id=Od5BxTEN0VsC|page=62}} 62] |isbn=978-0-582-24694-2 |year=1998 |publisher=CRC Press}}</ref>
: The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions [[rapidly decreasing|decrease sufficiently rapidly]] to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with [[Michel Plancherel|Plancherel's]] pathbreaking ''L''<sup>2</sup>-theory (1910), continuing with [[Norbert Wiener|Wiener's]] and [[Salomon Bochner|Bochner's]] works (around 1930) and culminating with the amalgamation into [[Laurent Schwartz|L. Schwartz's]] theory of [[Distribution (mathematics)|distributions]] (1945)&nbsp;...",<ref name=Kracht>{{cite book |title=Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A.L. Cauchy |chapter-url={{google books |plainurl=y |id=xIsPrSiDlZIC}}|first1=Manfred |last1=Kracht |first2=Erwin |last2=Kreyszig|author2-link=Erwin Kreyszig |page={{google books |plainurl=y |id=xIsPrSiDlZIC|page=553 }} 553] |isbn=978-9971-5-0666-7 |editor=Themistocles M. Rassias |year=1989 |publisher=World Scientific |chapter=On singular integral operators and generalizations}}</ref> and leading to the formal development of the Dirac delta function.