Help with proof for Dirca Delta Relation

[orion141]

I cannot think of how to go about proving this relationship and was wondering if any of you could help me


$$</font>\\int^{\\infty}_{-\\infty} f(x) \\delta \\left( g \\left( x \\right) \\right) dx = \\int^{\\infty}_{-\\infty} f(x) \\sum_{i} \\frac{\\delta \\left( x - x_{i} \\right)}{\\left| g' \\left( x_{i} \\right) \\right|} dx,<font color=red>$$

Thanks
Tom

 

[orion141]

I am not too good with this latex stuff so let me try again...

$$\\int^{\\infty}_{-\\infty} f(x) \\delta \\left( g \\left( x \\right) \\right) dx = \\int^{\\infty}_{-\\infty} f(x) \\sum_{i} \\frac{\\delta \\left( x - x_{i} \\right)}{\\left| g' \\left( x_{i} \\right) \\right|} dx$$

 

[orion141]

Well, that did not work either so I will link the relationship I am trying to prove. It is the dirac delta scaling relationship at the bottom of the page of this link.
https://www.physicsforums.com/showthread.php?t=122783&highlight=dirac+delta+scaling

 

[Rach3]

(i) Can you do the special case $$\delta(c x + b)$$?

(ii) And what happens to higher order terms in the Taylor expansion of g(x) near a zero g(x)=0, as arguments of the Dirac delta?

Use these things:
http://en.wikipedia.org/wiki/Dirac_delta_function#Representations_of_the_delta_function