Section 6.1 Dirac Delta Function
The Dirac delta function or simply, delta function or impulse function, is a mathematical function that is defined to be zero everywhere except at a single point, usually at the origin, where it is infinitely tall and integrates to one over an infinitesimally small interval. In order to deal with an impulsive force, which is a very large force acting for a short interval of time, and that of a load acting at a point of a beam which generates a very large pressure over a very small area, Dirac introduced this function in mechanics, called the Dirac Delta Function. This function is utilized to express physical quantities that are defined as a distinct point in time or in space as quantities which are distributed throughout time or space. This function is widely used in quantum mechanics and boundary values problems in mathematical physics.
The Dirac delta function is denoted by
\begin{equation}
\delta(x) = \begin{cases}
0, \quad \text{if} \quad x \neq 0\\
\infty, \quad \text{if} \quad x = 0
\end{cases}\\\tag{6.1.1}
\end{equation}
together with the condition that
\begin{equation*}
\int\limits_{-\infty}^{\infty}\delta(x)\,dx = 1
\end{equation*}
It is clear that \(\delta(x)\) is a function which is very large in the neighbourhood of the point \(x=0\) but is zero outside a very small interval surrounding the origin. \(\delta(x)\) may vary in this interval in an arbitrary manner without making its oscillations indefinitely large.
