How to represent DiracDelta?

Question

I need to represent δ(t-t0).
The hint is: In cases involving Dirac delta functions, onemay use the regularized delta function δ(t) = ε/[π(t^2 + ε^2)] approaching δ(t) in the limit ε → 0^+.
But I don't know how to insert that limit in Mathematica or represent the function in a different way.

Andreas · Answer

The delta_function(t-t0) is used as an operator  multiplied by some other function  g(t) inside a definite  integral over all t from plus to minus infinity and maps the function g(t) to a specific value g(t0) determined by the zero argument of the delta_function. To illustrate this look for example with a function t Cos[t] at
Integrate[(e/(Pi*((t - t0)^2 + e^2)))*(t*Cos[t]), {t, -Infinity, Infinity}]

giving some lengthy output containing a few expressions containing Floor functions . Determine their value for typical parameter values like
Floor[(Pi + 2*Arg[e + I*t0])/(4*Pi)] /. {e -> 10^(-12), t0 -> 5}

and enter their values into the lengthy integration result . Then simplify to get in this case :
(t0*Cos[t0] - e*Sin[t0])/E^e

Take the limit e -> 0 and the test function evaluated at t = t0 remains .

How to represent DiracDelta?

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