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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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    https://brainmass.com/math/graphs-and-functions/delta-function-formulas-446922

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    Solution:

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

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

    Solution Summary

    A delta function formula for limits by duality is examined.

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