Inner product of dirac delta function

[leroyjenkens]

Homework Statement

Find the inner product of f(x) = σ(x-x₀) and g(x) = cos(x)

Homework Equations

∫f(x)*g(x)dx

Limits of integration are -∞ to ∞

The Attempt at a Solution

First of all, what is the complex conjugate of σ(x-x₀)? Is it just σ(x-x₀)?

And I'm not sure how to multiply f(x) and g(x) together. Do I just pick out a value of x₀ that will make the dirac delta disappear and be left with the cos(x₀)?
Thanks

 

[strangerep]

What is ##\sigma(x-x_0)##? Did you mean ##\delta(x-x_0)##?

If so, the problem involves nothing more than the definition of the Dirac delta distribution. So perhaps you should write down that definition here, since it seems to be missing from your "relevant equations" section.

 

[leroyjenkens]

What is ##\sigma(x-x_0)##? Did you mean ##\delta(x-x_0)##?

If so, the problem involves nothing more than the definition of the Dirac delta distribution. So perhaps you should write down that definition here, since it seems to be missing from your "relevant equations" section.

edit: Oh, I was using sigma and I was supposed to be using delta; hence the "delta" function. They look similar.

The dirac delta function just picks out the value of x that makes x - x₀ = 0, and in that case, the function equals 1.

So basically, this problem is just a normal integral, since there's no complex conjugate of the dirac delta function?

Thanks.

 

[strangerep]

The dirac delta function just picks out the value of x that makes x - x₀ = 0,

That's the effect of the dirac delta when integrated against another function.

and in that case, the function equals 1.

I don't know which "function" you're referring to here, but I'm reasonably sure you still have some misconceptions about the Dirac delta (since you're still calling it a "function", or so it seems).

Take a look at the Dirac delta Wiki page. In particular the section titled "As a distribution". The distinction between "distribution" and "ordinary function" is very important for doing anything nontrivial with the Dirac delta.

So basically, this problem is just a normal integral

For simple cases, you can get away with thinking of it like that. But for less trivial stuff, you must understand the concept of "distribution" mentioned above.

there's no complex conjugate of the dirac delta function?

It's incorrect to say there's "no complex conjugate". Rather the Dirac delta is its own conjugate (like a real number).

 

[paranormal]

for your question. I would like to clarify a few things before providing a response.

Firstly, the inner product is typically defined for functions in a vector space, where the functions are considered as vectors and the inner product is a way to measure the angle between these vectors. However, in this context, it seems like we are dealing with the inner product in the context of Hilbert spaces, where the inner product is defined for functions but also has a more general definition.

Now, coming to the problem at hand, the inner product of two functions f(x) and g(x) in the context of Hilbert spaces is defined as ∫f(x)*g(x)dx, where the limits of integration are from -∞ to ∞. In this case, the functions f(x) and g(x) are σ(x-x0) and cos(x), respectively.

To answer your first question, the complex conjugate of σ(x-x0) would be σ*(x-x0). This is because the Dirac delta function is considered as a distribution and its complex conjugate is defined in terms of its action on test functions.

As for multiplying f(x) and g(x) together, we do not pick a value of x0 to make the Dirac delta function disappear. Instead, we use the properties of the Dirac delta function to simplify the calculation. In this case, we can use the property ∫f(x)*δ(x-x0)dx = f(x0) to simplify the inner product, as follows:

∫f(x)*g(x)dx = ∫σ(x-x0)*cos(x)dx
= ∫σ(x-x0)*cos(x)*δ(x-x0)dx (using the property above)
= cos(x0)*σ(x-x0)*∫δ(x-x0)dx (using the linearity of the integral)
= cos(x0)*σ(x-x0)*1 (since ∫δ(x-x0)dx = 1 by definition of the Dirac delta function)
= cos(x0)*σ(x-x0)

Therefore, the inner product of f(x) and g(x) is cos(x0)*σ(x-x0). I hope this helps to clarify the problem and the approach to solving it.