Mathematics

12:01 AM

@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

ccorn

@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 :-)

Semiclassical

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

ccorn

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

Semiclassical

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

ccorn

"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)

Semiclassical

12:18 AM

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

Jasper Loy

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

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

ccorn

@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.

Semiclassical

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

ccorn

12:35 AM

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 $\tau$.

Semiclassical

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 $k$

ccorn

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

Committing to a challenge

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

Semiclassical

right

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

Committing to a challenge

@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.

ccorn

12:46 AM

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.

Semiclassical

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

ccorn

"level" was meant as a technical term here

Semiclassical

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

ccorn

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.

Semiclassical

good point

Salehen Rahman

3:17 AM

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.

anon

3:58 AM

@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 ${\rm Iso}^+(\Bbb H)={\rm PSL}_2(\Bbb R)$.

Ryker

4:16 AM

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

The Substitute

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 $\phi: [a,b] \to \mathbb{R}$

anon

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

The Substitute

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

anon

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

The Substitute

yes, i got it from here. thanks

Chantry Cargill

4:50 AM

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

The Substitute

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?

anon

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

Ice Boy

5:09 AM

@JasperLoy neither

Martin Sleziak

5:22 AM

@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

Chantry Cargill

5:36 AM

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

UserX

6:06 AM

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

Martin Sleziak

@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.

2

Ice Boy

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

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

thanks for the link

Huy

6:51 AM

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

Martin Sleziak

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

Huy

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.

Martin Sleziak

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.

Ice Boy

here is the line :-)

Sawarnik

Aren't we talking of Europe?

Ice Boy

7:01 AM

@Sawarnik yes, I'm making a joke pal

Sawarnik

ok

@Huy Why did your gravatar change?

Huy

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

Ice Boy

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

Huy

7:18 AM

@IceBoy: Who knows?

Ice Boy

7:46 AM

or cares...

:)

UserX

8:05 AM

I log in with facebook and it synchronizes my avatar

Can I migrate a facebook account to an SE account?

The Artist

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?

The Substitute

10:43 AM

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?

Transcript for

$Mathematics$ Mathematics