Dirac Delta Integral Homework: Proving Equations

[iamalexalright]

Homework Statement

For some reason these are just messing me up. I need to prove:

1. \delta(y)=\delta(-y)

2.\delta^{'}(y) = -\delta^{'}(-y)

3.\delta(ay) = (1/a)\delta(y)

In 2, those are supposed to be first derivatives of the delta functions

Homework Equations

Use an integral with a test function

The Attempt at a Solution

Need a small hint

 

[Dick]

You need to prove the integral of delta(y)*f(y)*dy (what is that?) is equal to the integral of delta(-y)*f(y)*dy where y goes from -infinity to infinity and f is a test function. Try a change of variable u=(-y) and use the properties of the delta function. Don't forget to pay attention to what happens to the limits of integration when you change the variable.