I've encountered two different notations for the delta distribution: $$\delta(x,y)$$ and $$\delta(x-y)$$ What is the difference between these two notations. Do they depend on context or should I prefer one over the other?

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    $\begingroup$ The first one is not sufficiently standard for me to even know the intended meaning. The second one is quite standard (if the variable of integration is $x$, then this can be understood as a point mass of $1$ at $y$). $\endgroup$ – Ian Mar 29 '17 at 17:33
  • $\begingroup$ If the domain / space isn't additive, then $x-y$ makes no sense. But usually one writes $\delta_y$ if the mass is centered at $y$. $\endgroup$ – user251257 Mar 29 '17 at 17:38

The first describes a delta function acting at the origin.


The second describes a delta function acting along the line $x=y$:



The first notation is more convenient when working in high dimensions, since it naturally generalizes to $n$ dimensions: e.g., $\delta (x,y,z,w, ...)$. You can do this with the second notation $\delta(x) \delta(y), ...$

Certain software (e.g., Mathematica) supports both usages.

  • $\begingroup$ Thanks for your help with the Mathematica graphics. $\endgroup$ – dantopa Mar 29 '17 at 23:59

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