Editing Laplace transform (section)

== History ==
[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|Pierre-Simon, marquis de Laplace]]
The Laplace transform is named after [[mathematician]] and [[astronomer]] [[Pierre-Simon Laplace|Pierre-Simon, Marquis de Laplace]], who used a similar transform in his work on [[probability theory]].<ref>{{citation |url=https://archive.org/details/thorieanalytiqu01laplgoog |title=Théorie analytique des Probabilités |location=Paris |date=1814 |edition=2nd |at=chap.I sect.2-20 |chapter=Des Fonctions génératrices |trans-title=Analytical Probability Theory |trans-chapter=On generating functions |language=fr}}</ref> Laplace wrote extensively about the use of [[generating function]]s  (1814), and the integral form of the Laplace transform evolved naturally as a result.<ref>{{Cite book|title=Probability theory : the logic of science|last=Jaynes, E. T. (Edwin T.)|date=2003|publisher=Cambridge University Press|others=Bretthorst, G. Larry|isbn=0511065892|location=Cambridge, UK|oclc=57254076}}</ref>

Laplace's use of generating functions was similar to what is now known as the [[z-transform]], and he gave little attention to the [[continuous variable]] case which was discussed by [[Niels Henrik Abel]].<ref>{{citation |first=Niels H. |last=Abel|author-link=Niels Henrik Abel |chapter=Sur les fonctions génératrices et leurs déterminantes |date=1820 |title=Œuvres Complètes |language=fr |publication-date=1839 |volume=II |pages=77–88}} [https://books.google.com/books?id=6FtDAQAAMAAJ&pg=RA2-PA67 1881 edition]</ref> 

From 1744, [[Leonhard Euler]] investigated integrals of the form
<math display=block> z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx</math>
as solutions of differential equations, introducing in particular the [[gamma function]].<ref>{{harvnb|Euler|1744}}, {{harvnb|Euler|1753}}, {{harvnb|Euler|1769}}</ref> [[Joseph-Louis Lagrange]] was an admirer of Euler and, in his work on integrating [[probability density function]]s, investigated expressions of the form
<math display=block> \int X(x) e^{- a x } a^x\, dx,</math>
which resembles a Laplace transform.<ref>{{harvnb|Lagrange|1773}}</ref><ref>{{harvnb|Grattan-Guinness| 1997|p=260}}</ref>

These types of integrals seem first to have attracted Laplace's attention in 1782, where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.<ref>{{harvnb|Grattan-Guinness|1997|p=261}}</ref> However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form
<math display=block> \int x^s \varphi (x)\, dx,</math>
akin to a [[Mellin transform]], to transform the whole of a [[difference equation]], in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.<ref>{{harvnb|Grattan-Guinness|1997|pp=261–262}}</ref>

Laplace also recognised that [[Joseph Fourier]]'s method of [[Fourier series]] for solving the [[diffusion equation]] could only apply to a limited region of space, because those solutions were [[Periodic function|periodic]]. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.<ref>{{harvnb|Grattan-Guinness|1997|pp=262&ndash;266}}</ref>  In 1821, [[Cauchy]] developed an [[operational calculus]] for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering.  This method was popularized, and perhaps rediscovered, by [[Oliver Heaviside]] around the turn of the century.<ref>{{citation |first=Oliver |last=Heaviside |author-link=Oliver Heaviside |chapter=The solution of definite integrals by differential transformation |title=Electromagnetic Theory |location=London |at=section 526 |volume=III |chapter-url=https://books.google.com/books?id=y9auR0L6ZRcC&pg=PA234|isbn=9781605206189 |date=January 2008 }}</ref>

[[Bernhard Riemann]] used the Laplace transform in his 1859 paper ''[[On the number of primes less than a given magnitude]]'', in which he also developed the inversion theorem.  Riemann used the Laplace transform to develop the functional equation of the [[Riemann zeta function]], and this method{{clarify|reason=what method?|date=April 2025}} is still used to relate the [[modular form|modular transformation law]] of the [[Jacobi theta function]] to the functional equation{{clarify|reason=which functional equation?|date=April 2025}} .  

[[Hjalmar Mellin]] was among the first to study the Laplace transform, rigorously in the [[Karl Weierstrass]] school of analysis, and apply it to the study of [[differential equations]] and [[special functions]], at the turn of the 20th century.<ref>{{citation |first1=Murray F. |last1=Gardner |first2=John L. |last2=Barnes |title=Transients in Linear Systems studied by the Laplace Transform |date=1942 |location=New York |publisher=Wiley}}, Appendix C</ref>  At around the same time, Heaviside was busy with his operational calculus.  [[Thomas Joannes Stieltjes]] considered a generalization of the Laplace transform connected to his [[Stieltjes moment problem|work on moments]].  Other contributors in this time period included [[Mathias Lerch]],<ref>{{citation |first=Mathias |last=Lerch |author-link=Mathias Lerch |title=Sur un point de la théorie des fonctions génératrices d'Abel |journal=[[Acta Mathematica]] |volume=27 |date=1903 |pages=339–351 |doi=10.1007/BF02421315 |trans-title=Proof of the inversion formula |language=fr|doi-access=free |hdl=10338.dmlcz/501554 |hdl-access=free }}</ref> [[Oliver Heaviside]], and [[Thomas John I'Anson Bromwich|Thomas Bromwich]].<ref>{{citation |first=Thomas J. |last=Bromwich |author-link=Thomas John I'Anson Bromwich |title=Normal coordinates in dynamical systems |journal=[[Proceedings of the London Mathematical Society]] |volume=15 |pages=401–448 |date=1916 |doi=10.1112/plms/s2-15.1.401|url=https://zenodo.org/record/2319588 }}</ref>

In 1929, [[Vannevar Bush]] and [[Norbert Wiener]] published ''Operational Circuit Analysis'' as a text for engineering analysis of electrical circuits, applying both Fourier transforms and operational calculus, and in which they included one of the first predecessors of the modern table of Laplace transforms.
In 1934, [[Raymond Paley]] and [[Norbert Wiener]] published the important work ''Fourier transforms in the complex domain'', about what is now called the Laplace transform (see below).   Also during the 30s, the Laplace transform was instrumental in [[G H Hardy]] and [[John Edensor Littlewood]]'s study of [[tauberian theorem]]s, and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion.  [[Edward Charles Titchmarsh]] wrote the influential ''Introduction to the theory of the Fourier integral'' (1937).

The current widespread use of the transform (mainly in engineering) came about during and soon after [[World War II]],<ref>An influential book was: {{citation |first1=Murray F. |last1=Gardner |first2=John L. |last2=Barnes |title=Transients in Linear Systems studied by the Laplace Transform |date=1942 |location=New York |publisher=Wiley}}</ref> replacing the earlier Heaviside [[operational calculus]]. The advantages of the Laplace transform had been emphasized by [[Gustav Doetsch]],<ref>{{citation |first=Gustav |last=Doetsch |title=Theorie und Anwendung der Laplacesche Transformation |location=Berlin |date=1937 |publisher=Springer |language=de |trans-title=Theory and Application of the Laplace Transform}} translation 1943</ref> to whom the name Laplace transform is apparently due.