Dirac delta questions

Question

I'd like to ask for help in solving 3 I suppose quite easy questions about Dirac delta.

1. $$\delta (-x) = \delta (x)$$
2. $$\delta(ax) = \frac{1}{|a|}\delta (x)$$
3. $$\delta ' (-x) = -\delta ' (x)$$

Thanks in advance!!!

Answer 1

I'm assuming you are using the physicist definition of the Dirac delta distribution, where: $$\int_{-\infty}^{\infty} f(x)\delta(x)dx = f(0)$$ If this is the case, then the properties follow from:

1) $\int_{-\infty}^{\infty} f(x) \delta(-x)dx = -\int_{\infty}^{-\infty} f(-y)\delta(y)dy =\int_{-\infty}^{\infty}f(y)\delta(y)dy$

2) $\int_{-\infty}^{\infty}f(x)\delta(ax)dx = \frac{1}{a}\int_{-\infty}^{\infty}f\bigg{(}\frac{y}{a}\bigg{)}\delta(y)dy = \frac{1}{a}f(0) = \frac{1}{a}\int_{-\infty}^{\infty}f(x)\delta(x)$

3) $\int_{-\infty}^{\infty}f(x)\delta'(-x)dx = -\int_{-\infty}^{\infty}f'(x)\delta(-x)dx =-\int_{\infty}^{-\infty}(f')(-x))\delta(x)dx = \int_{-\infty}^{\infty}(f')(-x)\delta(x)dx =f'(0) = \int_{-\infty}^{\infty}f'(x)\delta(x)dx = \int_{-\infty}^{\infty}f(x)(-\delta'(x))$