Simply Beautiful Art's realm of calculus and analysis

« first day (169 days earlier) ← previous day next day → last day (230 days later) »

Simply Beautiful Art

20:11

@trying Hello and welcome to my realm!

I do stuff, so yeah

trying

@Simply Beautiful Hi, thanks

I'll bear it in mind.

Simply Beautiful Art

@trying What sorts of math are you interested in?

trying

fundamentals: abstract algebra, general topology, representation theory just to start and then functional Analysis, algebraic geometry, Dynamical system

Simply Beautiful Art

@trying Xam does abstract algebra and occasionally checks in my other realm. He's quite lonely lol

trying

I don't think to be able to answer math questions in real time in a chat. Probably on the forum is simpler for me. I need a little time to elaborate answers.

... and I don't have need at the moment to ask questions

I'm a self-learner

Simply Beautiful Art

20:21

@trying That is fine. Actually, you don't have to participate real time in chat, pinging usually works fine, and my chats (usually) don't get too busy, so messages aren't lost

@trying Cool

trying

ok, I need to become acquainted with the chat. Today it has been the first time for me

Simply Beautiful Art

@trying for the most part, I just do analysis. I've done some googology, which is the study and nomenclature of extremely large finite numbers, if you happen to be interested

By the way, do you happen to know how to get the MathJax working in chat?

trying

For what I need yes

Simply Beautiful Art

Okay :)

trying

googlology seems to be too special for my need. As I told you I need to learn the classic math.

Simply Beautiful Art

20:25

@trying for self interest I presume?

and yeah, actually, I don't think googology has many practical applications. Way too specialized and recreational

trying

It is a need (mine) of unkonwn origin

I'm not searching "pratical applications", anyway.

Simply Beautiful Art

:-) I think it is mostly the same for any who study math out of their own enjoyment

Also, @user21820 in my other realm does logic stuff

trying

although I think they must be at the beginning of every math study

Simply Beautiful Art

@trying Practical as in "can be applied to any other field of math"

@trying Personally, Euler's formula was at the beginning of why I started to study math

trying

My philosophy is very near to that of Vladimir Arnold

Good to know :-)

OK I gonna have a dinner now...

It was nice meeting you.

Simply Beautiful Art

20:30

Cya, nice meeting you too :D

trying

Ciao

Simply Beautiful Art

Zai jian

@Nilknarf Psst, no accepted answers: math.stackexchange.com/questions/2354004/…

2 hours later…

Nilknarf

23:03

@Simply Hey!

I can now access mod tools! :D

@SimplyBeautifulArt And I have a fun problem for you. You've probably seen it, but if not, you might find it interesting:
$$\int_{0}^1 \bigg\{\frac{1}{\sqrt{x}}\bigg\}dx$$

Simply Beautiful Art

Hey @Nilknarf

Nilknarf

Where $\{\}$ is fractional part

Simply Beautiful Art

@Nilknarf I have seen that, though I don't remember the solution off the top of my head. Probably involves splitting it into floor functions and then turning the integral into a summation

Nilknarf

Yeah, pretty much. No need to use floors though, it doesn't get that gross.

Simply Beautiful Art

But it should be doable that way...

Nilknarf

23:07

There is one problem that I need to ask your help with though:
$$\int_0^{1/\pi} \sin\bigg(\frac{1}{x}\bigg) dx$$

Simply Beautiful Art

@Nilknarf You might be interested in this question

@Nilknarf Easy, set $x=1/t$

Nilknarf

I managed to boil it down to
$$\int_\pi^\infty \frac{\sin(x)dx}{x^2}$$

Simply Beautiful Art

$$\frac{\sin(x)}{x^2}=\Im\left(\frac{e^{ix}}{x^2}\right)$$

$$\int_\pi^\infty\frac{e^{ix}}{x^2}~\mathrm dx =\lim_{t\to0^+}\int_\pi^\infty \frac{e^{ix-tx}}{x^2}~\mathrm dt$$

Nilknarf

But that uses Exponential Integrals D:

Simply Beautiful Art

Consider $d/dt$ of that last integral

and it comes out nicely

(may have to apply $d^2/dt^2$)

Nilknarf

23:11

So...

I have to simultaneously integrate and differentiate or something?

Simply Beautiful Art

Nah, you could alternatively use$$\frac1x=\int_0^\infty e^{-xt}~\mathrm dt$$

Nilknarf

You make it look so easy

Simply Beautiful Art

$$\frac{\sin(x)}{x^2}= \int_0^\infty\int_0^\infty e^{-xt}e^{-xu}\sin(x)~\mathrm dt~\mathrm du$$

This can easily be integrated w.r.t. $x$

Nilknarf

Argh, these mod powers are making me anxious

Simply Beautiful Art

Nilknarf

23:13

This bothers me

Simply Beautiful Art

Same actually

though I'm a bit busy atm lol

@Nilknarf learn yourself some common integral transforms mate

brb

Nilknarf

Any recommendations for where I can learn those?

My calc book doesn't cover them

@SimplyBeautifulArt @amWhy Also, close as dupe? People are answering it even though it has a duplicate:
https://math.stackexchange.com/questions/2405185/the-answer-to-the-equation-log-3x-log-2x-log-4x-1

Simply Beautiful Art

@Nilknarf drive.google.com/file/d/0BxKdOVsjsuEwdjBEM1dpRkhMa2s/view

Many integral and series stuff, doesn't provide step-by-step, but general layout, very useful resource.

Nilknarf

Oh, goody. Bookmarked!

Simply Beautiful Art

By the way, the pdf is from Jack D'Aurizio ;)

@Nilknarf it is also on his profile page

Nilknarf

23:24

"From"? Did he give it to you, or did he make it?

Simply Beautiful Art