# Delta function formulas

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?

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

#### Solution Preview

see attachment

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.