Two proofs in Dirac Delta Function

[Shing]

Homework Statement

a.)
Given $$\delta_n=\frac{ne^{-{n^2}{x^2}}}{\pi}$$
Show: $$x{\frac{d}{dt}\delta_n}=-\delta_n$$

b.)
For the finite interval $(\pi,-\pi)$ expand the dirac delta function $\delta(x-t)$ in sines and cosines, sinnx, cosnx, n=1,2,3... They are not orthogonal, they are normalized to unity (btw, what meant by "normalized to unity"?)

The Attempt at a Solution

a.)
I first claim that$\frac{d}{dt}\delta=0$ is always zero, as a single jumping point can be ignored(?) when it comes to slope.
and then I start with $\frac{d}{dt}(x\delta)=0$
However, I doubt
1.) is the slope truly equal to zero when x=0?
2.) How precise, rigor the math should be given it is a Math Method course?

b.)
I have totally no idea of this one! All I know is that it is related to Fourier transform, but I, we all haven't learned any about it yet!

Thanks for reading!

 

[CompuChip]

Note that the delta in question a) is not the Dirac delta which, informally speaking, is "zero everywhere except in 0."
Rather, they are defining a whole set of (differentiable) functions,
$$f_n(x) = \frac{n e^{-n^2 x^2}}{\pi}$$
and asking you to verify that
$$x \frac{\mathrm d}{\mathrm dt} f_n(x) = - f_n(x)$$
for all values of n, which you can do by simply plugging in the definition (hopefully switching the letters in the notation clears up the confusion).

The idea, of course, is that we can later look at
$$\delta(x) = \lim_{n \to \infty} \delta_n(x)$$
which is a limit of differentiable functions, but it is not itself a function (if you ever learn about distributions: this is the textbook example of a distribution which is not a function).

For b), you have to find the numbers a_n and b_n such that
$$\delta(x) = \sum_{n = 0}^{\infty} (a_n \sin(n x) + b_n \cos(n x))$$
Have you ever heard of Fourier transformation?

 

[Shing]

Thanks for reply!
Later on I started with the definition of derivative of Dirac Delta function
$$\int{x \frac{\mathrm d}{\mathrm dt} f_n(x)dx} = \int{- f_n(x)}dx$$
and claim that $x \frac{\mathrm d}{\mathrm dt} f_n(x) = - f_n(x)$
is a consequence of it (because I couldn't directl prove that :( )
Is it okay for this proof as it is asking we to use the $\delta_n=\frac{ne^{-{n^2}{x^2}}}{\pi}$

yeah, I have heard of Fourier transformation, but only learned little about it on my Mechanics course, and I didn't really get it :(

 

[CompuChip]

Once more, a) has nothing to do with the Dirac delta function!
If I asked you: prove that f(x) = x² satisfies
$$x \frac{d}{dx} f(x) = \frac{1}{2} f(x)$$
you wouldn't need it either, would you?

Or, a better analog would be, to define a bunch of function as $f_n(x) = x^n$ and ask you to show that
$$x \frac{d}{dx} f_n(x) = \frac{1}{n} f(x)$$.

How would you go about doing that?

 

[Shing]

oh I see...
so the very idea here is just to prove a limit about infinite?

oh then it is hard /.\
okay
I will try my best.