Fourier Transform of Delta Function: Solving Homework Statement

[spaghetti3451]

Homework Statement

Show that \int^{\infty}_{-\infty} e^{-ipt} dt = \delta(t).

Homework Equations

The Attempt at a Solution

I know that I must Fourier transform \delta(t), but not sure how.

 

[HallsofIvy]

Homework Statement

Show that \int^{\infty}_{-\infty} e^{-ipt} dt = \delta(t).

You can't. It's not true. The expression on the left is a function of p only, the expression on the right is a function of t only. Now, what is the problem really?

Homework Equations

The Attempt at a Solution

I know that I must Fourier transform \delta(t), but not sure how.

 

[spaghetti3451]

Ok. My mistake: the delta function should be a function of p, not t.

 

[jbunniii]

Homework Statement

Show that \int^{\infty}_{-\infty} e^{-ipt} dt = \delta(t).

The problem is that this isn't actually true. The integral on the left side doesn't exist.

What is true is that the constant function 1 has a Fourier transform (in the sense of distributions) equal to a Dirac delta function, possibly with a 2 \pi scale factor. However, you need quite a lot of mathematical machinery even to talk about what "in the sense of distributions" means.

If this is for an engineering course where these technicalities don't matter, then argue as follows:

1. The Fourier transform of a Dirac delta function is a constant
2. Therefore the inverse Fourier transform of a constant must be a Dirac delta function
3. Interpret the integral above as the inverse Fourier transform of a constant.

You should be able to prove 1. quite easily using the sampling/sifting property of the delta function. For 2 and 3 you will have to wave your hands a lot, but that's what engineering textbooks do in this case, too.

 

[Herr Malus]

Might I suggest simply placing the delta function in place of f(x) in the formula for the Fourier Transform?