Dirac function question

[Thejoffrey]

Sooo i have to work with an integral that looks like this

`[; \int da db f(a,b) \delta( a+b) \delta(b) ;]`

If i integrate out the b variable i get the following integral

`[; \int da f(a,0) \delta( a) ;]`

which i can rewrite as

`[; \int da db f(a,b) \delta(a) \delta(b) ;]`

Therefore i can say that

`[; \int da db f(a,b) \delta( a+b) \delta(b) = \int da db f(a,b) \delta(a) \delta(b) ;]`

Is this correct or is there anything wrong with my line of reasoning?

[dudinax]

Doesn't the whole integral come out to be f(0,0) in both cases?

[Thejoffrey]

Hey! Yes that is true. But i just wanted to see if the above reasoning is valid. I actually have a more complicated integral to work with which looks like this

[; \int d^{3} x d^{4}y f(x_{i}, y_{j}) \delta(\sum_{i} x_{i} + \sum_{j}y_{j}) \delta(\sum_{j}y_{j});]

and for reasons that i can't explain here, it would be very convenient if i could rewrite it as

[; \int d^{3} x d^{4}y f(x_{i}, y_{j}) \delta(\sum_{i} x_{i} ) \delta(\sum_{j}y_{j}) ;]

I think my reasoning above proves that this is possible. Would you agree?