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'
\int^{infnty}_{-infnty} f(x)\delta(x-a)dx = f(a)
\delta(a(x-b))=\frac{\delta(x-b)}{|a|}
\delta(-x)=\delta(x)
H'(t)=\delta(t)
\delta(x)=\frac{1}{2\pi} + \frac{1}{\pi}(cos(x) + cos(2x) + ...)

Are there any dirac delta functions for derivative and complex? \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!