I am just writing some notes on Fourier transforms and wanted some proofs for the dirac delta function methods as we didn't prove them but want my notes to be 'complete'

$\displaystyle \int^{infnty}_{-infnty} f(x)\delta(x-a)dx = f(a)$

$\displaystyle \delta(a(x-b))=\frac{\delta(x-b)}{|a|}$

$\displaystyle \delta(-x)=\delta(x)$

$\displaystyle H'(t)=\delta(t)$

$\displaystyle \delta(x)=\frac{1}{2\pi} + \frac{1}{\pi}(cos(x) + cos(2x) + ...)$

Are there any dirac delta functions for derivative and complex? $\displaystyle \delta'(t) and \delta*(t) $ ?

Don't feel obliged to proof all if you know of any websites that give accurate derivations them please feel free to quote them - I've had a good look around and can't find any useful sources

Thanks!