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History

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:

f (x) = 1 2 π ∫ − ∞ ∞ d α f (α) ∫ − ∞ ∞ d p cos (p x − p α) , {\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha)\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha)\,}

which is tantamount to the introduction of the δ -function in the form:

δ (x − α) = 1 2 π ∫ − ∞ ∞ d p cos (p x − p α). {\displaystyle \delta (x-\alpha)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha)\.}

Later, Augustin Cauchy expressed the theorem using exponentials:

f (x) = 1 2 π ∫ − ∞ ∞ e i p x (∫ − ∞ ∞ e − i p α f (α) d α) d p. {\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha)\,d\alpha \right)\,dp.}

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).

As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ -function as

f (x) = 1 2 π ∫ − ∞ ∞ e i p x (∫ − ∞ ∞ e − i p α f (α) d α) d p = 1 2 π ∫ − ∞ ∞ (∫ − ∞ ∞ e i p x e − i p α d p) f (α) d α = ∫ − ∞ ∞ δ (x − α) f (α) d α, {\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha)\,d\alpha \right)\,dp\\&={\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{aligned}}}

where the δ -function is expressed as

δ (x − α) = 1 2 π ∫ − ∞ ∞ e i p (x − α) d p. {\displaystyle \delta (x-\alpha)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha)}\,dp\.}

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:

Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L -theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945)...", and leading to the formal development of the Dirac delta function.

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. 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 Gaussians, which also corresponded to Lord Kelvin 's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics and used in his textbook The Principles of Quantum Mechanics. He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.