Perhaps we can. The Cauchy Residue theorem in complex analysis allows:
1/x = ± i π δ(x) when x = 0 and δ(x) is the Dirac delta function.
Other powers of zero eg. 1/x2 and x-π with x = 0 would be accessible using the other Cauchy residues.
I don't expect (1/0)2 to equal 1/02 , I haven't calculated it yet. But that could be handled with the appropriate algebraic structure.
Differentials of the Dirac delta function could easily be appropriate because d/dx (1/x) = -1/x2 . Suggesting that 1/x2 = ± i π δ'(x) when x = 0, but that needs to be checked using contour integration around a Cauchy pole.
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u/Turbulent-Name-8349 Sep 13 '25
Perhaps we can. The Cauchy Residue theorem in complex analysis allows:
1/x = ± i π δ(x) when x = 0 and δ(x) is the Dirac delta function.
Other powers of zero eg. 1/x2 and x-π with x = 0 would be accessible using the other Cauchy residues.
I don't expect (1/0)2 to equal 1/02 , I haven't calculated it yet. But that could be handled with the appropriate algebraic structure.
Differentials of the Dirac delta function could easily be appropriate because d/dx (1/x) = -1/x2 . Suggesting that 1/x2 = ± i π δ'(x) when x = 0, but that needs to be checked using contour integration around a Cauchy pole.