Dirac’s Delta

Part of the UNITEXT for Physics book series (UNITEXTPH)


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 Green also implies the concept. Often the names are not historically fair) \(\theta \) discontinuous function, also known as the step function, defined by
$$\begin{aligned} \theta (x) = \left\{ \begin{array}{ll} 1 &{} \text {se } x>0, \\ {1 \over 2} &{} \text {if } x=0, \\ 0 &{} \text {se } x<0, \end{array} \right. \end{aligned}$$
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
$$\begin{aligned} \int _{-\infty }^{\infty }\delta _{\alpha }(x) d x=1. \end{aligned}$$

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Università di Roma Tor VergataRomeItaly

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