In the IBM, a smooth approximation  of Dirac's delta
It has been proved that the inverse operator is the solution of the related equation of the following form, in which the non-homogeneous term has been substituted by the Dirac's delta function :
It should be noted that the Dirac's delta function, because of its rather complex nature, is formally described in terms of the derivative of the Heaviside unit step function :
Diagrams of the Dirac's delta function and Heaviside unit step function are schematically shown in Fig.
Therefore, using Dirac's delta distribution functions, the singular forces must be written as equivalent distributed ones.
n](t) = 0; h(t-[tau]) is the response to Dirac's delta function [delta](t), and is given by
where [delta](S) is the Dirac's delta whose value is [infinity] when S = 0 (i.
Changing the integration sequence and performing the integration on y using the Dirac's delta
In electrical engineering and other disciplines, the output (response) of a (stationary, or time- or space-invariant) linear system is the convolution of the input (excitation) with the system's response to an impulse or Dirac's delta
The signals are represented by Dirac's delta
distribution, for which the weight is set by the shaded part of the bottom plot.
To express a very sharp peak in branching rate at a time t = [Tau], we may use Dirac's delta function [Delta](t - [Tau]):
This can be shown by considering the case in which the branching rate has a very sharp peak at a time t = [Tau], expressed in terms of Dirac's delta function [Delta](t - [Tau]):