Looking for additional material about limits and distributions

[mynick]

I would like some help to find some additional info on generalized functions, generalized limits. My aim is to understand the strict definition of delta dirac δ(τ).If you could provide a concise tutorial focusing on δ(τ) not the entire theory...it would be of great help. I am not a math person,i know basic calculus.Also, if you know other online resources like videos,websites etc on this topic it would be of great help.

I am studying the delta dirac function δ(τ). My book talks about a strict, mathy definition of the delta function. It is using a generalized function formula. It says that δ(τ) is a special case of a bigger family of generalized functions also known as distributions. ∫ f(t)x(t) dt = Nf[x(t)], the integral is from - infinity to + infinity. Then the book briefly proceeds in a few lines and describes generalized limits. lim∫ f(t)x(t) dt =∫ f(t)x(t) dt , where lim is n->infinity. It proceeds and says that limit fn(t)=f(t), again lim is n-> infinity. I am sorry i do not know how to insert math formulas.

 

[Mark44]

I would like some help to find some additional info on generalized functions, generalized limits. My aim is to understand the strict definition of delta dirac δ(τ).If you could provide a concise tutorial focusing on δ(τ) not the entire theory...it would be of great help. I am not a math person,i know basic calculus.Also, if you know other online resources like videos,websites etc on this topic it would be of great help.

Have you looked here? https://en.wikipedia.org/wiki/Dirac_delta_function
This page describes the Dirac delta function as "a function that is equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations that were done by the theoretical physicists appeared to mathematicians as nonsense, until the introduction of distributions by Laurent Schwartz..."
There are links on that page to distributions and generalized functions.

I am studying the delta dirac function δ(τ). My book talks about a strict, mathy definition of the delta function. It is using a generalized function formula. It says that δ(τ) is a special case of a bigger family of generalized functions also known as distributions. ∫ f(t)x(t) dt = Nf[x(t)], the integral is from - infinity to + infinity. Then the book briefly proceeds in a few lines and describes generalized limits. lim∫ f(t)x(t) dt =∫ f(t)x(t) dt , where lim is n->infinity.

Most likely in your formula above, you mean $$\lim_{n \to \infty} \int_{-\infty}^\infty f_n(t)x(t)dt = \int_{-\infty}^\infty f(t)x(t)dt$$
IOW the integral on the left involves a sequence of functions $\{f_n(t)\}$.

It proceeds and says that limit fn(t)=f(t), again lim is n-> infinity. I am sorry i do not know how to insert math formulas.

See our tutorial here: https://www.physicsforums.com/help/latexhelp/
The script I used for the formula above looks like this: \lim_{n \to \infty} \int_{-\infty}^\infty f_n(t)x(t)dt = \int_{-\infty}^\infty f(t)x(t)dt