Product of a delta function and functions of its arguments

[ELB27]

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 $\delta(x-x')$ is the Dirac delta function and $f,g$ 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 $x'$ 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 $h(x)$. 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!

 

[micromass]

I don't have an answer, but if you replace $y$ with any concrete number, then the equality becomes correct. So wolfram is interpreting $y$ as a variable, while in your proof you treat it as a constant.

 

[BvU]

Can someone give a counterexample ?

 

[Ray Vickson]

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 $\delta(x-x')$ is the Dirac delta function and $f,g$ 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 $x'$ 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 $h(x)$. 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!

For which special case did WolframAlpha say it is false? AFIK what you wrote is true, at least if $f$ and $g$ are "nice" functions.

If we understand that $x'$ is the integration variable, then the four operators
$$L_1 = f(x') g(x) \delta(x'-x)\\ L_2 = f(x) g(x') \delta(x'-x)\\ L_3 = f(x')g(x') \delta(x'-x)\\ L_4= f(x) g(x) \delta(x'-x)$$
produce the same result when applied to any test function $h(x')$, so in the generalized-function sense we have $L_1 = L_2 = L_3=L_4$.

I think it is important that $x'$ and $x$ are "independent"; if they somehow vary together (for example, if $x = \phi(x')$) then, of course, everything can change.

 

[ELB27]

For which special case did WolframAlpha say it is false? AFIK what you wrote is true, at least if $f$ and $g$ are "nice" functions.

If we understand that $x'$ is the integration variable, then the four operators
$$L_1 = f(x') g(x) \delta(x'-x)\\ L_2 = f(x) g(x') \delta(x'-x)\\ L_3 = f(x')g(x') \delta(x'-x)\\ L_4= f(x) g(x) \delta(x'-x)$$
produce the same result when applied to any test function $h(x')$, so in the generalized-function sense we have $L_1 = L_2 = L_3=L_4$.

I think it is important that $x'$ and $x$ are "independent"; if they somehow vary together (for example, if $x = \phi(x')$) then, of course, everything can change.

Thanks! I indeed meant that $x$ and $x'$ are independent. The special case I entered into WolframAlpha is $xy\delta (x-y) \overset{?}{=} x^2\delta (x-y)$ to which it returned false. Following micromass' comment, I entered $y=5$ and it returned true as an "alternate form" while still answering false as the main answer. Apparently something's wrong with WolframAlpha itself then.

Thanks again!

EDIT: As a follow up: To prove some equality of generalized functions, is it sufficient to simply show as above that they give the same result for all possible test functions? In other words, can such functions be distinguished in any way without considering their action on a test function?