@ccorn: Do you know a good reference for elliptic integrals/functions? I always wanted to have a better handle on the stuff you referenced in this answer

@Semiclassical Classics first: Whittaker & Watson: A course in modern analysis. You'll never regret it. It covers elliptic functions and theta functions. Then, apart from the more systematic textbooks, I have found Borwein & Borwein's "Pi and the AGM" extremely worthwhile. And Weber's Lehrbuch der Algebra, vol. III, if you can read german and know in advance what he wants to tell you :-)

i probably can't put off W&W, can I

hadn't seen the B&B before

and I can't read german :P

i have a slight ulterior motive, in that your answer to that question always left me wanting more

Once you want to look at the modular bits, look at Zagier's texts. I find them enlightening.

ok. i know the ODE side pretty well, but modular stuff kind've goes over my head

"In the theory of Theta functions it is easy to find an arbitrarily large number of relations, but the difficulty begins where the objective is to find a way out of the maze of formulae. Occupation with those masses of formulae seems to have a withering effect on the mathematical intuition." -- G. Frobenius (translation attempt by me)

that's about how i feel whenever i try to open a book on theta functions

Theta functions? I only know sine of theta, LOL.

Guys, should I use lol or LOL in chat? Hmm...

@Semiclassical weakly related, I'd like to read about a method that can symbolically invert and simplify power series (or infinite lower triangular toeplitz matrices), giving general formulae for the terms instead of just working out initial ones.

i'd need to see that elaborated to quite follow it

the reason it came up for me was the physical interpretation of certain complete elliptic integrals, and the desire to efficiently invert/relate them

hence why things like K'/K are interesting

In a sense, the elliptic realm is circular, so you can always arrive at an inverse by going forward. From a period ratio $\tau$ define the nome $q$, from that compute Theta functions, from that elliptic integrals $K$ and $K'$, and their ratio gives you back $\tau$.

that makes sense. unfortunately the maze of things, as you quoted earlier, makes it hard to see the forest for the trees

in my case, i'd ideally want everything in terms of the $k$

In the above spirit, $k$ is just a Theta quotient

@JasperLoy I wrote fare-well, since most people don't interpret it as 'I want you to fare well'.

right

which is a nice statement, but makes my head spin a bit when i try to make practical use of it

@Chris'ssis I haven't slept $12$ hours in my concious life(Since I was a baby etc). I haven't slept $10$ hours since I was very sick many years ago and I sleep more than $7$ hours very rarely.

The Frobenius quote above heads the preface of a book by Günther Köhler on "Eta products and Theta series identities". Apart from working out 600 pages of formulae for levels up to 100, it has a very well written introductory/background part. Like Borwein's "Pi and the AGM", it manages to tell you almost en passant how to do some seemingly awkward bits of modforms calculations easily.

the chess allusion seems like an apt one, in that it the amount of complexity makes more sense the more of the logic that's understood

"level" was meant as a technical term here

ah. one i'd have to see to appreciate, i suspect

Note however that such a catalogue will only collect dust in your bookshelf unless you get some symbolic calculater (e. g. Pari/GP) and try out the presented series representations. Then you are in for a string of "Wow! It really works out!" moments.

good point

Let's say I wanted to "phase" two values, $a_0$ and $b_0$, and derive two new values, such that $\text{atan2}(a, b) = \text{atan2}(a_0, b_0) + c$. Is there a quick way to do that? Initially, I was thinking of getting the magnitude, $r$, of $a_0$ and $b_0$, and then getting the $\theta = \text{atan2}(a_0, b_0)$, and then getting $a = sin(\theta + c)r$ and $b = cos(\theta + c)r$. That's a lot of steps. I'm hoping to cut a lot of those steps down.

@ccorn Another hyperbolic question. Letting $\Bbb E$ be the Euclidean plane, we know ${\rm Iso}^+(\Bbb E)$ (the group of orientation-preserving isometries) is ${\rm Aff}(\Bbb R^2)$. As it acts transitively, all point stabilizers are conjugate, wlog pick a point $p$. We can write ${\rm Iso}^+(\Bbb E)$ as an internal semidirect product $\Bbb R^2\rtimes{\rm Stab}(p)$. Now consider ${\rm Iso}^+(\Bbb H)$ and $p\in\Bbb H$; can we write it as a semidirect product $\square\rtimes{\rm Stab}(p)$?

Note that ${\rm Stab}(0)$ within ${\rm Iso}^+(\Bbb E)$ is ${\rm SO}_2(\Bbb R)$, and ${\rm Stab}(i)$ is ${\rm PSO}_2(\Bbb R)$ within ${\rm Iso}^+(\Bbb H)={\rm PSL}_2(\Bbb R)$.

Is there anyone here that's studying or teaching at a European university?

off topic: If I'm taking a limit $x \to 0$ of an integral and the quantity $x$ I am taking the limit of appears in the limits of integration (my limits are $a+x,b-x$ for some reals $a,b$) can I push the limit into the limits of integration? My function in the integrand is of the form $\phi: [a,b] \to \mathbb{R}$

Show that |int_(b-x)^(a+x) minus int_b^a| approaches 0 in the limit

use the fact that phi must be bounded, and basic properties of the integral

@anon are no other requirements needed on $\phi$? Does it matter if it is $C^1$?

integrable

are you trying to do what I told you to do?

well, hmm, only assuming integrable might make the exercise too difficult, you can just assume bounded and integrable, which is a little bit looser than continuous

yes, i got it from here. thanks

I was wondering if anyone here could recommend me a book on differential geometry.

off topic: If $\gamma: [a,b]\to \mathbb{C}$ is a curve in the plane with $\gamma = \gamma_1 + i*\gamma_2$, then if $\gamma$ is differentiable, the author of my text says "then we write $d/dt \gamma = d/dt \gamma_1" + i*d/dt \gamma_2$. Is this a definition of the derivative of a curve in the plane or does it follow because $d/dt$ is a linear operator?

follows, although there's a narrow possibility that author is using it as a definition for whatever reason

@JasperLoy neither

@Ryker I am teaching in Bratislava, Slovakia. But there are big differences between university systems in various countries. (Mainly there are differences between eastern and western Europe, although this is big simplification.) So it is quite probable that if you have some very specific question, I will not be able to answer you.

@ChantryCargill Book for Undergrad Differential Geometry on main

Have you tried looking at questions tagged differential-geometry+reference-request or differential-geometry+book-recommendation

@MartinSleziak Thank you. That was exactly what I was looking for.

@MartinSleziak what are the main big differences between eastern and western universities?

@UserX I don't think I can fully answer that. But, as far as I know, PhD study is organized somewhat differently here and in the western countries. (But it seems to be changing.)

In the west they seem to start with attending lectures and studying for some time and only after some exam they proceed to start working on the topic of PhD thesis.

Here the topic is chosen already at the beginning of the PhD study. An it is basically the main thing the PhD-student should do.

There are also differences between the way staff is organized, see for example some things mentioned here.

@MartinSleziak the vote distribution on those answers is interesting :)

notable exclusions would be the UK, Germany, and Russia :(

thanks for the link

@MartinSleziak: What do you consider west and what east?

@Huy Basically former Western and Eastern (communist) block.

It is not a well-defined map just by evaluating it at two arguments.

Ok.

PhD students over here don't have to attend lectures etc, before starting to work on the thesis. I consider Switzerland being part of western Europa.

But I definitely do not claim that I know details about how universities work in all countries from the former Eastern block.

ok, so you see that it works differently from what I thought.

here is the line :-)

Aren't we talking of Europe?

@Sawarnik yes, I'm making a joke pal

ok

@Huy Why did your gravatar change?

@Sawarnik: It's a mystery, really. You'd know if you knew my Facebook password.

What does facebook have to do with auto-generated avatars at Stack Exchange?

@IceBoy: Who knows?

or cares...

:)

I log in with facebook and it synchronizes my avatar

Can I migrate a facebook account to an SE account?

Hey people , I found something weird. Is there something called the reduced Column echelon form?

I asked a question on finding the rank of a matrix, and I received an answer which is reducing columns, by adding one column to another. How is this valid?

If I am taking a complex line integral over a circle containing some singularities of my function, is there a way to deal with the singularities other than using the residue theorem?

Greetings

@Committingtoachallenge I only hope the downvotes series does not come from you.

Greetings

@IceBoy How have you been?

@ParthKohli fine thanks, how are you?

I received downvotes for very nice questions like this one

This is a rare gem, hard to find on sites and books.

@Chris'ssis this is a community site. People downvote for various reasons. My best guess would be that you insulted them somehow in chat or something.

Although unbased according to the vote guidelines provided, there is the human parameter here.

@UserX No, I never insult people.

you can't please all of the people all of the time

You are not the most pleasant person either. I can assume you have pissed some people off

Is the example 4 at math24.net/differential-operators.html correct?

I think the differential operator polynomial is wrong

@UserX Do you think I wanna be the most pleasant person?

@Chris'ssis do you think I care about any attitude of anyone in an internet chat?

Rhetorical answer; I don't. But some people do.

@UserX Well, yeah, because I did the mistake of sharing some of my best achievements. I won't repeat this mistake and then I'll be careless.

@IceBoy: Why should it have any impact?

if used unwisely it could

@IceBoy Can it read my shitty handwriting and solve hard DEs and integrals?

dunno

@IceBoy: I don't understand. Please evaluate.

evaluate what?

@IceBoy: How could it impact math education if used unwisely?

that is the question

Link to the question?

I'm typing it up now.

Wow, the comment that Chris's Sis is narcissistic has 8 stars. This is absurd.

Define absurd lol

Chris's Sis is not narcissistic, full stop.

Please star the above instead.

@UserX Thanks for using my favourite lol, lol.

internet stars, like internet points mean nothing

HOW ON EARTH NOT TO GO CRAZY ...? Look at what this user writes to me now ...

@IceBoy: I think the answer should be "none" then.

ok

:-)))))))))))

@Chris'ssis What is the smile for, what I said?

@JasperLoy It is "Damn, I live in a so sick world ... just smile" :-)))))

don't worry, be happy

@Chris'ssis what point is Katie trying to make?

@Chris'ssis Katie says "Going through living alone, but Math makes me see the light." Wow, that sounds awesome.

And why the fuck did I see 3 complex number problems today on MSE, all easily factorable and reduced to quadratics done with polar/exponential forms? Why do people prefer that method while it's longer and easier to do a mistake with?

@UserX Welcome to MSE, where poor answers are upvoted and good answers are downvoted, lol.

@JasperLoy if it's a method they've seen before they upvote it, if it's a new method they didn't learn in school they downvote it. Isn't that retarded?

It should be the other way around.

@UserX Yes. This world is full of stupidity and evil. I am still struggling to make sense of it myself.

@Chris'ssis are you 12?

Lol

@IceBoy Thanks.

@UserX For Katie, yeah, only 12. :-)

@Chris'ssis how old are you? What are you researching? How come you meet so many interesting integrals in your research?

How was this created? i.sstatic.net/qxUqC.gif

@robjohn Here is a cute limit to take :D

@Chris'ssis I think I need to flag you then, you must be 13 to sign up at Stack Exchange.

@UserX It is well known in this chat that her age and name is a secret for now.

@UserX I wanna write a book about integrals, series and limits, so I study a lot of things.

@DanielFischer lol, do I? :-)

@DanielFischer where is that rule?

Does mental age count?

@Chris'ssis I would be delighted to read your book and the methods you use on integrals.

@UserX No, only the birth date.

not everything that can be counted counts

@IceBoy A WISE PERSON ^^

@DanielFischer that's too bad. So many people would get banned if mental age counted.

not everything that counts can be counted

Most people stop growing mentally at 18.

WAIT! Let's define what "growing mentally" is .. (some might undersand different things)

As many people age, instead of growing wiser, they become more foolish, because they learn all the wrong things from this sick world, instead of applying critical thinking skills to evaluate what they see.

@Chris'ssis he was :D

@JasperLoy That sounds correct!

@Iceboy I would like a source

@IceBoy I thought it was you, really! :-)))

nah

:D

@UserX use google

@robjohn I'll include that limit in my book

My father told me one of his professors died and now a professor of similar specialty advises 24 masters(he took every person the deceased one advised). Is this even legal?

why wouldn't it be legal?

Isn't there a maximum amount of persons one single professor can advise?

I don't think so.

It depends on the person.

Some take none.

is $A \times B = B^{T} \times A^{T}$ ?

@lvella No, but the transpose of the left is equal to the right.

TeX math should be rendered here?

thanks

@lvella See math.ucla.edu/~robjohn/math/mathjax.html for LaTeX in chat.

math.ucla.edu/~robjohn/math/mathjax.html

I wonder how one would react when finds such a requirement on one of the pages of a book

:-))))

Mathematica requires no pen or paper.

lol, true

I've almost finished reading Courant calculus, is it still necessary to read Apostol's Calculus, or one can directly go to Rudin after Courant ?

@XingdongZuo No need for Apostol.

@XingdongZuo Apostol's Calculus, but volume II is very useful.

@user91500 That is not answering his question.

@JasperLoy I don't know if he has enough time, so I'm agree with you :)

@user91500 You are from Germany? My favourite country! I hope to be reborn there my next life, lol.

Wow, this night feels so nice :)

@JasperLoy It's impossible, but you can assume that you are from germany :)

@JasperLoy Oh, why?

How is the cardinaliry of the set of primes the same as the cardinality of the set of natural numbers if the set of primes is a subset of $\Bbb N$

@UserX Both are countable. Infinite cardinalities don't work the same way as finite one.

@Sawarnik Too many reasons to list here.

@user91500 It is possible since I believe in reincarnation, lol.

@JasperLoy You can try a few.

@JasperLoy Why don't you come to Bihar then? :P

@Sawarnik It has good mathematics.

@Sawarnik I prefer living in places with liberal laws and attitudes.

@JasperLoy Is that grammatically correct?

@JasperLoy What do you mean? :/

@Sawarnik TLDR.

@Sawarnik Yes, if it means what I want it to mean.

@JasperLoy :?

@JasperLoy What do you mean then?

@Sawarnik Never mind. =) How is your school?

@JasperLoy Boring.

@Sawarnik Never mind. =)

@Sawarnik Are you going to IIT?

Never mind.

@JasperLoy Not sure :|

@Sawarnik Do you still talk to Rachel?

Most of my friends will, but I am not very sure.

@JasperLoy Rachel?

@Sawarnik math101, lol.

Oh, no.

@JasperLoy is $\Bbb C\setminus \Bbb R$ countable?

@UserX It is uncountable.

@JasperLoy How can I prove the set of primes to be countable?

@Sawarnik $\Bbb P \subset \Bbb N \implies |\Bbb P|=\Bbb N|=\aleph_0$

I guess

@Sawarnik It depends on what you mean by countable or proof. Wait till you do a course in set theory.

But you have to prove that every subset of $\Bbb N$ has the same cardinality of $\Bbb N$ and is thus countable by definition.

But that's sheer guesswork, I don't know if this is correcr.

Correct*

Every INFINITE subset*

@JasperLoy Check out that very lllllhf math.stackexchange.com/questions/987424/… :P

@TheGame Hi! It is a hhf, I am only a banana, lol.

@TheGame Check my limit above ...

@Chris'ssis Something is wrong there

@Chris'ssis $\max_{1\le n \le N}\{\sqrt{n}\}=\sqrt{N}$

@TheGame There you have the fractional part ...

@Chris'ssis Nooo :/ bad notation xD

@TheGame :D

Usually we write $\max_{index}\{\text{elements of a set}\}$

Conflicting notations

@Chris'ssis Wow it's even more interesting then. Just by looking at it it looks like it has no limit.

@TheGame Are you sure it has no limit?

as $1-\max_{1\le n \le N}\{\sqrt{n}\}$ is evolving slowly, but keeps getting all the values between $0$ and $1$ 'increasingly' as $N$ grows

@Chris'ssis I'm not sure, I'm just saying it's surprising

@TheGame Yeah, true.

@Chris'ssis Upvote my question above :D It looks horrible

@TheGame Done.

:c

@TheGame I upvoted it! :D

Yeah I know thanks :)

@TheGame Me too!

So many starred posts about you in the starred board :c @Chris'ssis

@JasperLoy ;D here a cupcake

I would upvote it too if I understood it

@TheGame I rather have a girlfriend, lol.

@UserX You can always upvote it now and understand later :P wicked logic

Hands over a girlfriend to JasperLoy

@TheGame if you explain it to me to the extent that I understand it I'll upvote it. However I'm way far from understanding it.

@TheGame I think I'm about to become a star too ... that's why ... :-))))))

@Chris'ssis :DDDDD

A narcicistic star. Sounds plausible.

@Chris'ssis You are already Ramanujan to me, lol.

@JasperLoy Thanks! The appreciation is rare these days ... :D

@TheGame

@MikeMiller You should post that as an answer.

@MikeMiller What does that have to do with 't small enough' ?

@TheGame Some of those series don't converge if $t$ is too big; it depends on the matrix norm of $f$.

:-)))

@UserX ^^

Well, I chose it at random, but it was about stars ... ;)

Anyway, let me write up the proof to that limit and then create another version.

@TheGame I'm going to add that limit to my book, I think it's a very nice limit, not like the usual ones.

@Chris'ssis I hope so ! :D

@TheGame Do you have a guess about the answer?

@Chris'ssis Will it be available for next Christmas ? :D

@TheGame Definitely, I hope so.

@Chris'ssis Not really :/

@Chris'ssis If it was $\ln$ instead of $\sqrt{}$ I might try $0$

@Chris'ssis Wouldn't the limit be the same if you replaced $\sqrt{}$ by any concave function with a limit $+\infty$ in $+\infty$ ?

@TheGame I didn't consider a generalization yet, but I have some ideas ...

away walking for 1 hour or so

@TheGame like jogging?

It's also interesting to consider $$\lim_{N\to\infty} \sqrt{N}\left(1-\max_{1\le n \le N}\{n-\sqrt{n}\} \right)$$

What might I say about it?

WAIT!!! (I just created a 3rd limit for my book)

$$\lim_{N\to\infty} \frac{1-\max_{1\le n \le N}\{\sqrt{n}\}}{1-\max_{1\le n \le N}\{n-\sqrt{n}\}}$$ it looks divinely!

@robjohn see my last version, it's simply absolutely amazing, crazy amazing.

Or I might write it as $$\lim_{N\to\infty} \frac{1-\max_{1\le n \le N}\{n-\sqrt{n}\}}{1-\max_{1\le n \le N}\{\sqrt{n}\}}$$

@Chris'ssis: Your book sounds expensive...

@Nick I only wanna have my book first, I don't think of other gains.

Can someone give me a line of the form $ax + by = c$ where $x$ and $y$ can never be integers simultaneously... or is that impossible?

@Chris'ssis: Promise me it will be more awesome than I imagine.

@Nick I don't know what kind of thing you may imagine. The imagination is hard to beat. ;)

@Chris'ssis: ... you'll beat it. I know you will :D Good Luck as always.

@Nick lol, thanks :-)

Nick dashes away like a jolly bowl of jelly to learn more calculus

Wait, I'm not done yet ...

Prove that $$\lim_{N\to\infty}\max{\left(\left( \frac{1-\max_{1\le n \le N}\{n-\sqrt{n}\}}{1-\max_{1\le n \le N}\{\sqrt{n}\}}\right)^{\sqrt{N}}\right)}=e$$

Why is that limit amazing?

@thisismuchhealthier Can you compute it?

Probably

@thisismuchhealthier note there is a fractional part.

Not today though, but tomorrow I am free

lol, tomorrow anyone can compute anything :D

What do you mean beautiful Nick

tomorrow never comes... you have infinite time to learn :D

@Nick LOL

Oh haha, very funny!

Nah on the 24th my time zone I will do

24th of october in 2014

Before you say something smart!

Ooh, good luck then, hope your DeLorean has enough plutonium to hit 88mph.

Otherwise, you're in big trouble.

Nah, forget it. The limit isn't that hard, But good luck, anyhow :D

Is $2^{\aleph_0}$ literally 2 raised to the power of $aleph_0$ or is it a notation convention?

@UserX: ... $2^{\aleph_0}$ gives me the shivers....

What are the shivers

@UserX the cardinality of hom(X,Y) (the set of maps X->Y) is |Y|^|X|. For finite |X| this is a fact, for infinite it is a definition of cardinal exponentiation. (in particular, the power set of X is equivalent to hom(X,2))

@Chris'ssis $\frac12$

@robjohn Yeah, you did it. :-)

@robjohn Did you try the second one? $$\lim_{N\to\infty} \sqrt{N}\left(1-\max_{1\le n \le N}\{n-\sqrt{n}\} \right)$$

@Chris'ssis That is also $\frac12$

@robjohn Yeap :D

back

@Chris'ssis The ratio is usually approximately $1-\frac1{2N}$, but when $N$ is a perfect square (or even one less than a perfect square), it jumps to approximately $1+\frac1{\sqrt{N}}$

@TheGame: to the future

@Chris'ssis I'm too lazy to jog :P

@Nick >:c

@Chris'ssis :OOOOOOO

@Chris'ssis WAAAAAAT

That is crazy !

@robjohn The limit can be either $1$ or $e$, but taking max we get $e$.

@Chris'ssis I figured you meant $\limsup$

@TheGame :D

@robjohn all his max are lim sup ? or ... ?

@Chris'ssis Owait it it maxs or lim sups

@robjohn Yeah, you're right. My notations fails once in a while. Still, is it wrong to consider max there?

@TheGame Well, it's better to use lim sup, it's far more natural ...

@Chris'ssis I updated my previous statement to show the ratio's size when less than $1$, and yes, raising to the $\sqrt{N}$ power would yield $1$ or $e$

@robjohn Exactly. :-)

These cookies are now ready to be served in some math contest and to be added to my book :D:D:D:D:D

@Chris'ssis There should be a way to pre-order your book >:c

:P

@TheGame :D

@TheGame: Get a plane ticket. Knock on her door. And beg. That's my plan.

@Nick hahaha :-)))

@Nick I don't have the address :c

@TheGame: ... Just follow the light

@robjohn The first limit wasn't created by me, but the rest ones. I received that first one by email from a kid.

@Nick :c What is that, a sect ? xD

Follow the Guru @Chris'ssis

Let me rewrite that last one $$\limsup_{N\to\infty}\left( \frac{1-\max_{1\le n \le N}\{n-\sqrt{n}\}}{1-\max_{1\le n \le N}\{\sqrt{n}\}}\right)^{\sqrt{N}}=e$$

@TheGame: No, I meant I'd be coming along with you with a flashlights and torches. We'll find her. We'll get pictures. We'll prove that she is real. Next, we'll search for Bigfoot and the Lochness Monster.

The readers of my book will love me totally! :D

@Nick Prove that she is real ?

@Nick Lolno i found those two already

@Nick They were playing chess with the Yeti

@TheGame: ... you know too much. Why has no one nueralized you yet?

@Nick Nueralize ? Have you been reading too much Inuyasha ?

Hello, all. Does anyone know a straight-forward, intuitive approach to learning about Lebesgue measures? Any sources would be awesome

@TheGame: Who is Inuyasha? I've never heard of that half demon before in my life, I swear.

-____-

@Anastasiya-Romanova would be happy xD

I am struggling with fundamental Euclidean geometry. Ugh.

Feels so silly.