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Now Dirac was writing in the 30's, back before they had a rigorous theory of distributions, so he was freely manipulating the Dirac delta. The identity he derived is certainly correct, but is adding a constant multiple of the delta function when you divide by x really mathematically justified? If it is, why isn't this technique more widely adopted, like when solving Laplace's equation or doing quantum mechanics, places where delta functions usually abound?

Any help would be greatly appreciated.

Thank You in Advance.