A "simple" application of dirac delta "shift theorem"...help. (FT.. integration..etc)
1. The problem statement, all variables and given/known data
show that for a, b, c, d positive:
δ(a/b-c/d) = bdδ(ad-bc)
2. Relevant equations
∫f(x)δ(x-a)dx = f(a)
3. The attempt at a solution
Ok so I start with
∫δ(a/b-c/d)f(x)dx
But I am not sure how to apply the shift theorem. It seems I need to somehow relate a/b and x so that I can get it in the form of the shift theorem. But trying integration by substitution I always get tangled up. If I let u=a/b, then I can't relate dx to du to intergrate.
If I just say let's call a/b as "x". then dx = what?
ugh, this is a simple problem too. Seems like its an easy canditate for one of the first proofs shown after learning about the shift theorem so I feel pretty dumb that I'm not sure even where to start...
1. The problem statement, all variables and given/known data
show that for a, b, c, d positive:
δ(a/b-c/d) = bdδ(ad-bc)
2. Relevant equations
∫f(x)δ(x-a)dx = f(a)
3. The attempt at a solution
Ok so I start with
∫δ(a/b-c/d)f(x)dx
But I am not sure how to apply the shift theorem. It seems I need to somehow relate a/b and x so that I can get it in the form of the shift theorem. But trying integration by substitution I always get tangled up. If I let u=a/b, then I can't relate dx to du to intergrate.
If I just say let's call a/b as "x". then dx = what?
ugh, this is a simple problem too. Seems like its an easy canditate for one of the first proofs shown after learning about the shift theorem so I feel pretty dumb that I'm not sure even where to start...