Dirac Distribution with 2 Diracs

[divB]

Hi,

It is well known that

$\int f(x) \delta(x-a) dx = f(a) \quad\mathrm{and}\\ \int f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a)$

Similarly, Is there a way to express a distribution integral with multiplicative Diracs in a compact form (e.g., a sum)?

$\int f(x) \delta(t-x-l_1) \delta(t-x-l_2) dx$

Thanks,
divB

 

[HallsofIvy]

The way you have written this, with a single x, is a bit misleading. What you show would be simply f(t- l) if $l_1= l_2= l$ and 0 if $l_1\ne l_2$.

But I suspect you mean a multi-dimensional version where x, t, and l are all vectors or shorthand for multiple variables. In that case,
$$\int\int f(x,y)\delta(t_x- x- l_1)\delta(t_y- y- l_2)dxdy= f(t_x-l_1, t_y- l_2)$$

 

[divB]

Thank you, your answer was exactly what I was looking for!