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Delta function formulas

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a) Justify the formula ?(2x)=(1/2)?(x) by limits and by duality;
b) Find a similar formula for ?(ax) when a>0 and when a<0;
c) We know that x?(x)=0, where ? is the delta function. On the other hand by Leibniz rule (x?(x))â?²=?(x)+x?â?²(x) is apparently not zero. How can this paradox be explained?

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a) Let's consider the following "delta" sequence:
( 1)
so that
( 2)
It can be checked that:

Replacing x with 2x, we will have:
( 3)
Since ...

Solution Summary

A delta function formula for limits by duality is examined.