Lighthill generalized function book

[rar0308]

Homework Statement

see the pictures

Homework Equations

The Attempt at a Solution

one.
In the third picture.
3rd line and 4th line. i don't understand why 1-e^(-t^2/n^2) become (1+t^2)/n^2
two.
I'm not familiar with proving equality by showing difference going to zero.
for example, I prove equality by showing that A equal B, B equal C, so A equal C.
What is a good book to learn this way of proof ?

 

[gopher_p]

Homework Statement

see the pictures
View attachment 66493View attachment 66494View attachment 66492

Homework Equations

The Attempt at a Solution

one.
In the third picture.
3rd line and 4th line. i don't understand why 1-e^(-t^2/n^2) become (1+t^2)/n^2

It's an inequality. $1-e^{-t^2/n^2}$ doesn't "become" $\frac{1+t^2}{n^2}$; they're saying that $1-e^{-t^2/n^2}\leq\frac{1+t^2}{n^2}$, which is provable using elementary calculus (hint: what is the global min of $g(t)=\frac{1+t^2}{n^2}+e^{-t^2/n^2}-1$?).

two.
I'm not familiar with proving equality by showing difference going to zero.
for example, I prove equality by showing that A equal B, B equal C, so A equal C.
What is a good book to learn this way of proof ?

This method of proof, using $\lim(f-a)=0$ to show that $\lim f=a$ (please pardon the awful notation), is very common in analysis. I wouldn't characterize it as something that is "learned"; you become aware of it (which you apparently have) and then you recognize when it's being used in other proofs and try to use it in your own proofs. I would say that it is comparable to using $a-b=0$ to show that $a=b$, which is something you probably wouldn't think twice about using (though someday you should ). As long as you're comfortable with the reasoning behind why it works, there's not much else to know about it. Just do it.

 

[rar0308]

Thanks.
your post is the answer.
What I always feel after seeing a solution of a problem is a regret that why I haven't come up with this idea.
Do you have some way to prevent this regret?

 

[gopher_p]

First off, just about any decent proof is obvious once you've seen it. But most of them aren't obvious at all until you've seen it. It's the whole "hindsight is 20/20" phenomenon. So don't feel bad if something that was out of reach suddenly becomes obvious; that's how it works. That's how it's supposed to work. It's completely normal.

This particular proof looks like it came from a textbook. Most textbook authors leave most proofs that are doable by their intended audience as exercises. The fact that this one was done for you is an indication that you maybe aren't supposed to be able to come up with the ideas on your own. Also keep in mind that the proof was most likely originally discovered by someone more experienced (and maybe even more intelligent *gasp*) than you and I, and it probably took them much longer than you think to come up with it.

Also, mathematics, especially at the higher levels, is a discipline that is practiced. You're not likely going to be good at it right off the bat, and no one really expects you to be. Just keep at it. The more work you do, the better you will be. The tricks/techniques only come with experience, and experience is, unfortunately, something that you can't rush. The good news is that you can learn a lot of math in a very short period of time if you just do it.