The Dirac delta function (sometimes called the unit impulse function) is important in the study of phenomena of an impulsive nature. The Green's Function is the impulse response of a differential equation, so the Dirac delta function has a central role in the method of Green's functions.
The Dirac delta function is defined to be zero when ,
and infinite at in such a way that the area under the function is
unity. A concise definition is the following: given non-zero numbers and ,
ASIDE: This is a ``weak'' definition of , since the limits of integration are never allowed to be precisely zero. This definition is sufficient for work with Green's functions. See Barton (1989, p. 11) for a discussion of ``weak'' and ``strong'' definitions.