- #1
ELB27
- 117
- 15
Homework Statement
I am trying to determine whether
$$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$
where [itex]\delta(x-x')[/itex] is the Dirac delta function and [itex] f,g[/itex] are some arbitrary (reasonably nice?) functions.
Homework Equations
The defining equation of a delta function:
$$\int_{-\infty}^{\infty} \delta(x-x')f(x')dx' = f(x)$$
(I'm supposing [itex]x'[/itex] is the variable of integration, but it shouldn't matter I think)
The Attempt at a Solution
It appears that from the integral definition of the delta function,
$$\int_{-\infty}^{\infty}f(x)g(x')\delta (x-x') h(x')dx' = f(x)g(x)h(x) = \int_{-\infty}^{\infty}f(x)g(x)\delta (x-x') h(x')dx' = \int_{-\infty}^{\infty}f(x')g(x')\delta (x-x') h(x')dx'$$
for all [itex]h(x)[/itex]. Thus, the above identity appears to be correct. However, when I ask WolframAlpha a special case of this question, the answer is that the identity is false. How can I determine which it is?
Any comments/suggestions would be highly appreciated!