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23:59
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Q: How to represent DiracDelta?

Marina NebotI 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.

plotting
user64494
user64494
This limit is not a usual limit, but the limit in the weak topology. Such limits are not imlemented in current CASes. All that is not a simple matter. See that Wiki article and/or Encyclopedia of Mathematics as a first reading. Something similar: an atom cannot be represented as a very small ball.
J. M.'s ennui
J. M.'s ennui
What will you be using DiracDelta[] for?
Marina Nebot
Marina Nebot
I'm not using DiracDelta for anything, I just want to represent it.
Leonid Shifrin
Leonid Shifrin
@MarinaNebot The limit you mentioned should be taken already after the delta-function is integrated over t in convolution with some other function. It only makes sense when you think of a delta function as a kernel of an integral operator. If you simply need to represent it somehow, why using the regularization and not the standard abstract delta-function notation (or, DiracDelta[] in WL)? Or, if you want to use the regularized version, simply denote it somehow and use that notation. Either way, you don't need to take the limit just for representation purposes, and it would make no sense.
user64494
user64494
@LeonidShifrin: The integral of the type $\int_{-\infty}^\infty\delta(x-y)f(y)\,dy$ makes no sense in math (see Generalized function to this end). That was noticed many times at the forum.
Leonid Shifrin
Leonid Shifrin
23:59
@user64494 You can always use the regularization and the limiting procedure to make sense of these kinds of integrals, understood as a limiting procedure rather than directly as a standard (Riemann) integral. From the practical viewpoint, what matters is the order of the limits, which is what I mentioned in my comment.
user64494
user64494
@LeonidShifrin: I prefer arguments over a tonn of emotional and ungrounded words (" From the practical viewpoint" etc).
Leonid Shifrin
Leonid Shifrin
@user64494 What I gave you is an argument. An (Riemann) integral is defined as a limiting procedure. You have two limits to take then. As long as we understand what the notation of delta function means - namely, a specific order of limits, integral first, and regularization last, there is no problem with the validity of the integral. I have spent a big chunk of my life working with delta functions in physics, first studying, then getting and publishing actual results. What I told you was what mattered in practice in my work. And yes, I am familiar with formal theory of generalized functions.
Roman
Roman
@LeonidShifrin you're feeding a troll. user64494 says "[...] That was noticed many times at the forum." Yes, by you and you alone. You spend so much time arguing against other people's use of the Dirac δ-function and yet refuse to study how it is actually used every day by millions of physicists/scientists. Has it occurred to you that maybe it's you who's mistaken here and in the dozens of other posts you complain about regularly?
anderstood
anderstood
The paragraph en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure might be useful. It's an abuse of notation that is to be understood in the context of Lebesgue integrals wrt the Dirac measure (and not the Lebesgue measure $dx$, and that's the abuse).
Roman
Roman
Does this answer your question? How to deal with a Dirac delta function numerically?
user64494
user64494
23:59
@LeonidShifrin : I looked at your current 672 answers at this forum and found only that on the topic.
@anderstood: The interpretation of the $\delta$-distribution as an atomic measure causes problems with ODEs of such type $y''(x)+y(x)=\delta(x-1)$.
Leonid Shifrin
Leonid Shifrin
@Roman "you're feeding a troll" - apparently yes, at least in the context of this Q/A. I was confused by their high reputation.
user64494
user64494
Colleagues, let us discuss topics, not persons. Do you understand me?
John Doty
John Doty
Are you talking about a graphical representation? A common convention is to use an upward pointing arrow. See en.wikipedia.org/wiki/Dirac_delta_function#/media/….

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