Dirac’s Delta

Let us start with the Heavyside (Oliver Heaviside (1850–1925) was probably the first to use the \(\delta \) before Dirac, and the work of George GreenDirac also implies the concept.Heaviside Often the names are not historically fair.) \(\theta \) discontinuous function, also known as the step function, defined by \(\begin{aligned} \theta (x) = \left\{ \begin{array}{ll} 1 & \text {for} x>0, \\ {1 \over 2} & \text {for } x=0, \\ 0 & \text { for } x<0. \end{array} \right. \end{aligned}\) With it we can define a rectangular-shaped peak function, of width \(2\alpha \) , \(\begin{aligned} \delta _{\alpha }(x) ={\theta (\alpha ^{2} - x^{2}) \over 2\alpha }, \end{aligned}\) such that \(\begin{aligned} \int _{-\infty }^{\infty }\delta _{\alpha }(x) d x=1. \end{aligned}\)