I want to define a specific operator that will act in generic functions that depend, say, in the variable z. My problem is that I want when a specific function is given as an input, to have specific outcome. For example, I would like to overload CircleTimes such that CircleTimes[f[z],g[z] will do nothing but if CircleTimes[f[z],DiracDelta[1-z]] the output to be f[1] and similarly for CircleTimes[DiracDelta[1-z],f[z]]. This should hold for any f[z].
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1 Answer
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Use TagSetDelayed (i.e. /: and :=) to associate a "special rule" to f only when it appears within CircleTimes with DiracDelta:
ClearAll[f]
f /: CircleTimes[f[_], DiracDelta[_]] := f[1]
f /: CircleTimes[DiracDelta[_], f[_]] := f[1]
You can then see the following:
CircleTimes[f[z + 1], DiracDelta[1 + x]] (* Out: f[1] *)
CircleTimes[DiracDelta[-z], f[whatever]] (* Out: f[1] *)
When a function that is not DiracDelta is involved, then no simplification is made:
CircleTimes[g[z], f[x]] (* Out: g[z] ⊗ f[x] *)
CircleTimes[f[z], g[z]] (* Out: f[z] ⊗ g[z] *)
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$\begingroup$ Thank you for your comment. Ok that looks good but this is only for function
f. I would like it to be for any functionfthat is dependent specifically on variable, say,z. That meansCircleTimes[c1 f[z],c2 DiracDelta[1-z]]=c1 c2 f[1].. $\endgroup$– halCommented Mar 7, 2019 at 14:58 -
$\begingroup$ @hal OK. It would have been much more helpful if you had specified those requirements directly in the original question though, so the question doesn't become a moving target $\endgroup$– MarcoBCommented Mar 7, 2019 at 15:02
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$\begingroup$ Thank you for your answer. I had. I stated that it should hold for any
f[z]. $\endgroup$– halCommented Mar 7, 2019 at 15:20 -
$\begingroup$ @hal Yes, you said for any
f[z]which here does NOT mean "any function of $z$", but any function of the formf[z]. We deal with code, not math. Also, note that in your exampleCircleTimes[c1 f[z],c2 DiracDelta[1-z]]=c1 c2 f[1]you are not specifying whatc1orc2should be. Are those numbers? Reals? Perhaps the easiest way is for you to give a few examples of expressions you expect to encounter, carefully definining what each variable means, together with the output you expect. Add those to your question to get better answers. $\endgroup$– MarcoBCommented Mar 7, 2019 at 15:25
CircleTimes[]commutative? Is the behavior ofCircleTimes[DiracDelta[1 - z], f[z]]unique toDiracDelta[1 - z], or should similar behavior be expected for e.g.DiracDelta[z - 3]? $\endgroup$DiracDelta[1-z]is enough. And yes it is commutative. $\endgroup$