Editing Dirac delta function (section)

====Oscillatory integrals====
In areas of physics such as [[wave propagation]] and [[wave|wave mechanics]], the equations involved are [[hyperbolic partial differential equations|hyperbolic]] and so may have more singular solutions.  As a result, the nascent delta functions that arise as fundamental solutions of the associated [[Cauchy problem]]s are generally [[oscillatory integral]]s.  An example, which comes from a solution of the [[Euler–Tricomi equation]] of [[transonic]] [[gas dynamics]],{{sfn|Vallée|Soares|2004|loc=§7.2}} is the rescaled [[Airy function]]
<math display="block">\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). </math>

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense.  Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the [[Dirichlet kernel]] below), rather than in the sense of measures.

Another example is the Cauchy problem for the [[wave equation]] in {{math|'''R'''<sup>1+1</sup>}}:{{sfn|Hörmander|1983|loc=§7.8}}
<math display="block"> \begin{align}
c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\
u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0.
\end{align} </math>

The solution {{mvar|u}} represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the [[sinc function]] (used widely in electronics and telecommunications)
<math display="block">\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk </math>

and the [[Bessel function]]
<math display="block">  \eta_\varepsilon(x) =  \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>