Mathematics - 2012-12-02 (page 1 of 7)

@OldJohn, yes. I agree. It just seems, for some reason, Calculus is particularly pyramid-like. I feel as though every problem pulls from a different bag of techniques.

Old John

@Limitless I felt like that with potential theory - the bags of tricks were: calculus, measure theory, complex analysis, topology, ...

Limitless

@OldJohn, what's potential theory?

Old John

12:08 AM

yeeha! 6K reached :)

@Limitless study of harmonic and subharmonic functions, mainly

- the only bit of maths I am really qualified to talk about,:)

user19161

Wow @amwhy only 200 more to go to 10k for you! =)

Limitless

@OldJohn, are the harmonic and subharmonic functions related to the harmonic numbers?

Old John

@Limitless possibly - but I don't know about harmonic numbers - only about harmonic functions

amWhy

@WillHunting Yeah...who knew? I can reach there by tomorrow's close of day, if I keep my streak going...

@WillHunting You're climbing pretty quickly, too!

user19161

@amWhy Yup! ... I added ellipses just for fun...

user19161

12:12 AM

@OldJohn Well done!

amWhy

@OldJohn Congrats!

Limitless

@OldJohn, I found harmonic numbers (and their generalizations) fascinating. They are the discrete calculus analog of pathological integrals.

Old John

@will @amWhy thanks!

Limitless

@amWhy, congrats on your progress. You too, John.

amWhy

@Limitless Thanks!

@Limitless You're doing well. I like your input!

Old John

12:15 AM

I should go to bed before I fall asleep at my laptop - or speak more rubbish

g'night all!

Limitless

@amWhy, lately I think I have been making significant progress. Thanks.

@OldJohn, bye! Thanks for everything!

Old John

@Limitless speak again - bye for now

user19161

@OldJohn Good night!

amWhy

I'm going to need to eat something (today)! I'll be around...

user19161

@amWhy Have a huge one!

Limitless

12:18 AM

@amWhy, how does this look so far? I've been having fun with it, but I think I have given enough help. I might post a full solution in a week, but I dunno.

user19161

@Limitless It is OK, but like I said a little misleading as the integral involving D and E should not be split up this way but in the way I suggested above.

amWhy

@Limitless Looks very good! +1!

Limitless

@WillHunting, that would be applicable if I knew specifically what $D$ and $E$ are.

user19161

@Limitless The first fraction in the split up should be of the form f'(x)/f(x) so that the antiderivative is ln |f(x)|.

Limitless

@WillHunting, nonetheless, I see your point and I think you're right.

@amWhy, thanks. I'm looking for more calculus questions right now.

Gustavo Bandeira

12:29 AM

@WillHunting Yeah. It's going to start on day 3, but this day is going to be only a magnum class - (I'm trying to translate it to english)

It's only a presentation of the university.

peoplepower

12:54 AM

@anon Yes, since if $p-1=4m$ then the permutation $\pi$ decomposes into pairs of $\varphi(d)$-cycles, a 2-cycle, and a set of $2\varphi(d)$-cycles where $d|m$. Whereas the only divisors of $4m+2$ are odd numbers and 2 easily proving $\pi$ is even.

user19161

1:06 AM

Hey @amwhy which distro do you run in your virtual box?

user1230219

Hi, I recently read about kepler's laws (for like the third time), and I realized that I should be able to get a better understanding of how newton confirmed them with my current knowledge of calculus. I understand newton's 3 laws (and his law for gravity), and kepler's three as well - I am having trouble starting, however, with turning newton's laws and moving masses into those elliptical orbits that I need. Any help of where I should start at would be greatly appreciated.

user19161

@Limitless Looking where?

Limitless

@WillHunting, here. Surprisingly

user19161

@Limitless Have you read the notes I suggested to you on Paul's site? I think you should work through those first.

KaliMa

what is the rotation of an ellipse?

user1230219

1:19 AM

@KaliMa, was that directed toward me?

KaliMa

anyone

an ellipse of form x^2+4y^2=4a^2 rotated counterclockwise by angle 90<theta<0

user1230219

I'm guessing that you want to translate that into polar form, and back to cartesian, or if you have already learned, use the equations for translation.

KaliMa

is there an easy way to just translate it cartesian?

Limitless

@WillHunting, I have read some of them. Not nearly enough. I am currently floating around. I just posted this.

I thought that question was helpful since I was getting rusty on solving basic recursions.

user1230219

well, you do have those two equations, y=r sin th, and x=r cos th...

user19161

1:23 AM

@Limitless Oh man, what a long answer. It's good but I think you don't need to give so many details to these askers you know.

Limitless

@WillHunting, I write with the hopes that I've made my process obvious. I really don't like Gauss's approach to mathematics. He once stated something along the lines of, "If I included an explanation of anything, my proofs would become tomes in themselves."

user19161

@Limitless I feel the same about some of my answers here. =) I mean, the Gauss part.

Limitless

@WillHunting, I think ca . . . did not understand his own statements in mathematical form.

@all, is this question even answerable as stated?

user19161

1:40 AM

@Limitless Not sure what the question means.

Limitless

@WillHunting, that's what I mean. I'm not sure either.

user1230219

I understand that third question

he's asking about questions in that form. A distinct problem would be like "10 distinct people are going to sit in a round table; how many ways can they do this"? etc

Limitless

1:56 AM

@user1230219, the question above assumes there are 10 seats rather than any other natural number?

peoplepower

...

user1230219

yes, sorry, that is the kind of question that he is asking about - his question is vague but I'm 90% sure he's trying to understand those questions @limitless

Limitless

@user1230219, I'd say his question is too broad, then.

his or her

user1230219

I disagree - honestly I see a kid trying to complete his/her math homework. And I see that kid trying to be good, and instead of posting his homework problems, he posts general problems that would help him solve and undersatnd what he needs to do.
I mean, yes, the questions are phrased vaguely and not very well, but they are much better than they could be.

Limitless

2:13 AM

@user1230219 You've got a point. Hm. Seems like this is a rough territory to judge.

Peter Sheldrick

hi, in graph theory, is a single isolated node its own clique?

or is a clique at least of size 2?

(homework tag)

KaliMa

how would i find all integral distances from the origin to an ellipse x^2+4y^2=4a^2?

Limitless

@KaliMa, $\frac{x^2}{4a^2}+\frac{y^2}{a^2}=1$. This ellipse is centered at the origin, so isn't the value you're looking for simply the distance from the origin of any given point $(u,v)$ on the ellipse? Namely, $\sqrt{(u-0)^2+(v-0)^2}=\sqrt{u^2+v^2}$. Given that $y(x)=\sqrt{a^2-\frac{x^2}{4}}$, we then have what for $v=y(u)$ and thusly $\sqrt{u^2+v^2}$? (Or am I simply missing the complexity of this question?)

KaliMa

i am looking for distances from the origin to any point on the elliptical, where those distances are integral

(u^2+v^2)^.5 gives the distance but it isn't necessarily integral

Limitless

@KaliMa, ah! I see now. So, finding the integral values of $f(u)=\sqrt{u^2+a^2-\frac{u^2}{4}}$?

KaliMa

2:27 AM

not sure what that is

Limitless

@KaliMa, that's what I wrote above. It's $\sqrt{u^2+v^2}$ with the $v^2$ simplified to $a^2-\frac{u^2}{4}$.

KaliMa

oh i see

u^2 + a^2 - v^2/4 = n^2 ?

for integer n

Limitless

What I get is the following:
$$
\begin{align}
\sqrt{u^2+v^2}&=\sqrt{u^2+y(u)^2}\\
&=\sqrt{u^2+\left(\sqrt{a^2-\frac{u^2}{4}} \right)^2}\\
&=\sqrt{u^2+a^2-\frac{u^2}{4}}\\
&=\sqrt{\frac{4u^2+4a^2-u^2}{4}}\\
&=\sqrt{\frac{3u^2+4a^2}{4}}\\
&=\frac{1}{2}\sqrt{3u^2+4a^2}.
\end{align}
$$

Thus, we have that $f(u)=\frac{1}{2}\sqrt{3u^2+4a^2}$ is integral if and only if $\sqrt{3u^2+4a^2}$ is divisible by $2$.

KaliMa

so 3u^2 + 4a^2 = 4n^2

Limitless

Yep, that is right.

So we know it's from the sequence $0, 4, 16, 36, \dots$.

KaliMa

2:35 AM

?

Limitless

The expression $3u^2+4a$ is in the sequence $\{4n^2\}_{n \in \mathbb{Z}}=\{0, 4, 16, 36,\dots\}$.

It's just a cool fact.

KaliMa

for a=209 I was able to find distances 152 and 190

Limitless

For $a=209$, you have $n=\frac{1}{2}\sqrt{3u^2+174724}$. There's a lot of possible distances, but I can't know how many there are, I think.

KaliMa

er sorry I mean 247 and 266

a=209
for n in range(a+1,2*a):
    threeU=4*n*n-4*a*a
    if threeU>=0:
        u=(threeU/float(3))**.5
        if int(u)==u:
            print n

i start n at a+1 and stop at 2a because the distance to the ellipse has to be at least the length of the semiminor and less than the semimajor, right?

Limitless

@KaliMa, if I knew programming, I could follow you. Sadly, I do not.

KaliMa

2:43 AM

i am basically looking at the problem here tinyurl.com/bq3bl4h

it seems like a good first idea is to determine integral lengths to the unrotated ellipse

Limitless

I dislike Project Euler. Nonetheless, I'll give it a stare for about 10 minutes.

Peter Sheldrick

hmm where does this dislike originate from?

KaliMa

triplet (a,b,c) pulls the $a$ term from $x^2+4y^2=4a^2$ and then $b$ and $c$ represent the integral distances to the intersections

Limitless

@KaliMa, ouch. My worst understanding is within conics, and this just about threw me through a loop.

@PeterSheldrick, I don't know how to program and all of the problems have this program-esque feel to them.

KaliMa

so what i am trying to do first is get an idea for which points on the ellipse have integral distances

(209, 247, 286) means that when a=209 the shortest integral distance to the ellipse is 247 and the other is 286

Peter Sheldrick

2:48 AM

@Limitless 'I don't know how to program' hmm too many proofs by contradiction? :P

Limitless

@PeterSheldrick, More of an axiom or a vacuous truth.

@KaliMa, from my point of view, this problem is mindboggling with the amount of variables.

From a purely abstract point of view, I have absolutely no feel for it.

KaliMa

well it's the same as before

just finding integral distances

using the equation you came up with though i don't get 247 for example to work

@Limitless any reason why 247 works as a distance with your equation but not 286?

@Limitless (1/2)sqrt(3*152^2+4*209^2) = 247

amWhy

3:11 AM

Hey Will Hunting, I just stopped back, myself!

Limitless

@KaliMa, I don't think you can just put the triplets into my equation.

@amWhy, can you help KaliMa out?

KaliMa

but you should be able to, i just derived the same equation myself

the 247 works, but not the 286, not sure why

user19161

@amWhy Hi there! So how was your dinner?

amWhy

@WillHunting Nothing to "rave" about: yogurt and cereal which I ate right from the box...and some milk. I'm craving "real food", like pizza!

user19161

@amWhy Hmm, I think that not wanting to cook is not a good reason to go vege. First, a lot of veg needs to be cooked. Second, some meat is easy to cook. QED.

amWhy

3:17 AM

@WillHunting You're right! hehehe. Very nice proof!

@WillHunting I used to cook, a lot. But I live alone, and sometimes it just doesn't seem worth the trouble to cook "just for me."

@WillHunting Interesting...a year or so ago, there was a post (quickly closed) asking whether there is a correlation between "thin-ness" and being a mathematician!

user19161

@amWhy I know of no great mathematician who is fat!

user19161

@amWhy Hmm make sure you get proper meals though!

amWhy

@WillHunting That was exactly what the questioner observed!

@WillHunting Yes, I need to be better about adding variety, and treating myself occasionally to things I crave.

Argon

Hi all!

KaliMa

Would anyone be able to help me with this ellipse

Limitless

3:26 AM

@WillHunting, I don't think it's possible to be good at maths and particularly fat. I think overweight is possible, but very unlikely. There is just way, way, way, way too much standing at blackboards, whiteboards, windows, in showers, . . . etc involved.

Argon

Hahahaha!

Limitless

@Argon, have you used The Art of Problem Solving? I am considering it.

Argon

@Limitless I have never used it, but I've read some interesting stuff there.

Limitless

@Argon, I'm referring to the two volumes. I hear it's directed at people like us (young, i.e. 13-17, people with an interest in mathematics).

Argon

@N3buchadnezzar I evaluated $$\int_0^1 \log x \log(1-x)\, dx$$

@Limitless Nope never used them. I will look into it more; what have you heard?

Jayesh Badwaik

3:30 AM

@WillHunting Poisson was plump.

Limitless

@Argon, my friend says they are useful, but not for the competitions he participates in as they are a tad bit above the level of the competitions he participates in. I participate in the same competitions, but I don't care about whether they'll be useful specifically for those competitions because I think any mathematical learning is good.

Argon

@JayeshBadwaik Poisson was a fish. Schmaltz herring!

Jayesh Badwaik

@Argon Hahahahahaha. ROFL

Write the stuff in the same line, then I can star it.

Argon

@Limitless Hmmm... Interesting!

Limitless

@JayeshBadwaik, wasn't Jacobi also large?

Argon

3:32 AM

I also do some contests, though not religiously.

Jayesh Badwaik

@WillHunting The notification has also gone.

@Limitless I do not know. I know about Poisson though.

Argon

@Limitless Do you like integrals?

Limitless

@Argon, I think the fine line is this: Learn mathematics, ignoring whether the competitions will pat you on the back. However, try not to ignore when you are being ranked below your competitors significantly. (That has yet to occur, thankfully.)

2

Argon

@Limitless :) Me too

Limitless

@Argon, they are growing on me. The area they possess in my heart is slowly increasing.

Argon

3:34 AM

@Limitless They are so elegant and mysterious!

Limitless

@Argon, are they not?!

Argon

They are awesome

Limitless

I cry over the fact that not all integrals have a closed form in terms of elementary functions.

Jayesh Badwaik

@Limitless That sums it all quite well.

Limitless

I had to get over the depression that made me feel.

Argon

3:35 AM

@Limitless That is their beauty!

Limitless

@Argon ???? Do elucidate!

Argon

@JayeshBadwaik Sums are fun too! Hahaha!

Limitless

@Argon, I recommend discrete calculus for sums.

See Concrete Mathematics. (Many of my answers cover topics in that book.)

Argon

@Limitless You can find definite integrals over regions, even when no elementary antiderivative exists!

Using many awesome ways!

Limitless

@Argon, I'm aware! That is awesome, now that I think about it. It reminds me of things such as the Dirac delta.

Jayesh Badwaik

3:37 AM

@Argon May be. As long as they sort themselves out. :P

Gotta go now. Bye.

Argon

@JayeshBadwaik I remember us trying to figure out $$\sum_{n=2}^\infty \frac{\log n}{n!}$$

Hahaha!

@JayeshBadwaik Bye!

Limitless

@JayeshBadwaik, bye!

Jayesh Badwaik

@Argon Yeah, that went nowhere. Nice Bernoulli connection though.

@Limitless Bye.

@Argon Bye.

Limitless

@Argon, you will love this. (I hope this isn't shameless self promotion; it's more like Knuthian love.)

Argon

@Limitless Also, complicated looking integrals may be simple to evaluate, while conversley, simple integrals may be hard!

"In Concrete Mathematics..." Hahaha!

Limitless

3:40 AM

@Argon, that reminds me of.. hm... There was this one integral that involved one single function and had the most confusing antiderivative I'd ever saw.

Argon

@Limitless Do you remember it?

Limitless

@Argon, it was a trippy integral involving something similar to the \erf function.

Argon

Hahahaha!

How about one of my favourites, $$\int_0^1 x^x\, dx$$

@Limitless Nice answer!!!! Cool tricks, +1

Limitless

Ah! Yes. The canonical $\int x^{x^{-2}}dx$.

Argon

@Limitless !!!!!

What's that?

$$\int x^{x^{-2}}\, dx$$

Limitless

3:43 AM

@Argon, thanks. The $\sum_i i^n$ set of sums is fascinating. And there's not an expression in terms of elementary functions. See wolframalpha.com/input/?i=integral+of+x%5E%28x%5E%28-2%29%29 .

Argon

@Limitless Oh, that's what you meant.

Check out this awesomeness:

$$\int_0^1 x^{-x}\, dx = \sum_{n=1}^\infty x^{-x}$$

Limitless

@Argon, oi. Isn't that related to Sophomore's Dream?

Argon

Thanks to Johann Bernoulli.

@Limitless Yep

Limitless

Oh. That is Sophomore's Dream.

@Argon, it is so rare to find people my age who are this interested in mathematics.

Argon

@Limitless Indeed. I personally have met no one else.

Limitless

3:47 AM

@Argon, I have met one. However, he's certainly not as interested as I am. He doesn't want to approach things like Abstract Algebra or really deep theory like the stuff behind why there are no general formulas for quintics and higher.

Argon

@Limitless I've always wanted to start abstract algebra. Where to start?

Limitless

@Argon, Lose all confidence in your mathematical ability first.

Argon

(Galois theory is baoss)

@Limitless HAAHAHA!

@Limitless Do you know complex analysis? I LOVE that stuff.

Limitless

@Argon, I started with the originator: van der Waerden. Algebra, by van der Waerden. Many professors and teachers have told me that wasn't a wise decision.

Argon

@Limitless Why not?

Limitless

3:49 AM

I thought it was very useful, but I hit a road block along chapter 3. Also, Algebra is over 50 years old. It's ancient.

Argon

@Limitless Whittaker and Watson is almost a century old, but still respected!

Limitless

Basic Algebra I by Jacobson is now what I use. It is much more up to date. However, I have yet to get past the introduction.

@Argon, Abstract Algebra is the type of field---based on my experiences---that gets newer motivations and perspectives much quicker than a lot of fields. Many things mentioned in van der Waerden's work have been viewed completely differently, since his work was the very basis of Abstract Algebra (teaching wise---he basically formalized lecture notes into the first book on Abstract Algebra).

Argon

@Limitless It's much more modern then the analysis that I'm familiar with. It seems really cool

Limitless

Also, category theory. Back in van der Waerden's time, this was completely unknown. Basic Algebra I uses a tid bit of it and introduces it when it's relevant.

Argon

@Limitless What will you do in university?

Limitless

3:53 AM

@Argon, pursue two Ph.Ds as of right now.

Argon

@Limitless In what subjects? :)

Limitless

@Argon, I bet you already know the first. Mathematics and Psychiatry.

Argon

@Limitless Wasn't expecting psychiatry!

Limitless

@Argon, I employ mathematics and my mathematical thinking to psychiatry often.

Argon

Maybe I should get an easy Ph.D in ergonomics (haha!) or something so people have to call me doctor.

@Limitless Really? Like for what?

Limitless

3:55 AM

I've also studied psychiatry longer. I started studying in 6th grade for psychiatry and in 8th grade for mathematics.

@Argon, you're Dr. regardless of whether your Ph.D is in medicine, as far as I know. And, let me think of an example . . .

Argon

@Limitless I don't think ergonomics count as medicine...

Hahahaa!

Limitless

OW, handcramp

Argon

Hahahaha!

user1230219

Hi, has anyone learned about Tycho Brahe?
I'm searching for the data that he supposedly spent his lifetime collecting, and I can't find it...

Argon

We had to do assignments on ergonomics in computers. Such a joke, hahaha

Limitless

3:59 AM

@Argon, one way I think of it is like this: People behave in ways that are modeled mathematically. For example, the stock market is modeled mathematically. However, on a deeper level, I think that many psychological characteristics (such as personality traits) are similar to mathematics. In higher mathematics, we see how many definitions build together to bring theorems forth. This is manifested in people by seeing how many events build personality traits. [. . .]

Furthermore, just as a field is much like (but not entirely) its collection of theorems, a person's personality is the collection of their personality traits.

@user1230219, I do not think his data survived.

Argon

@Limitless Hmmm.... Interesting. I never really had that neo-Pythagorean way of thinking about stuff like that.

user1230219

Well i found the name >.<
Rudolphine Tables

Argon

hmmm

user1230219

Then I found the link- and realized that I am ignorant and they aren't in English >.<

Limitless

@Argon, a more specific example is this: All human emotions are like functions of time, and many mental illnesses are like special functions.

For example, I see bi polar disorder as a trigonometric function.

Argon

4:01 AM

@Limitless Hahahaha! But they seldom can be described in such a predictable way, no?

(forgive me for my ignorance of the subject)

Limitless

@Argon, but they can! And that's the beauty of my thinking. I often know what will happen based on logical prediction.

These mathematical analogies help me make a more thorough understanding of what I already know and I discover new things by viewing them in a different light.

Argon

@Limitless But, with, say, bi-polar disorder as a periodic function, what defines the "period?"

It's not as clear cut as saying "$2\pi$"

Limitless

@Argon, wonderful question. Let's be a bit more specific: We're saying that the nature of the emotion is modeled by a trigonometric function. That is, positive values mean positive emotions and negative values mean negative emotions. The period would correlate roughly to the average time it takes the person to experience an extreme low and return to an extreme low after experiencing an extreme high.

Argon

@Limitless And this can be generalized like this? It can be used to make future predictions?

Limitless

@Argon, yes. Not very specific, but yes. For example, if we know that someone's period is 2 months, we know that we should ask them about ideas at the end of the two months. Since they are feeling negative, they will give us an inaccurate opinion and therefore note all the negative aspects of our idea (which may or may not exist). Likewise, we should wait until the middle of those 2 months to ask again and receive all of the positive aspects of our idea.

Argon

4:07 AM

Hmm, interesting. Started your Ph.D thesis already, I see :)

Limitless

@Argon, not at all. Just brainstorming. :-)

I'm also a little versed in philosophy. So, I often throw that into the mix too.

Argon

You seem quite well rounded then!

Limitless

@Argon, I have had to be! Social skills are quite hard to come by. :P

Argon

@Limitless Hahahah! Very true

Limitless

Speaking of Jacobson, I really would like to get back into Abstract Algebra. It is simply divergent with my interests right now. Another analogy: Convincing someone to do something is like a convergent or divergent series where events are the terms of the series.

@Argon, and I know absolutely no complex analysis. I'm terrible, I know.

Argon

4:11 AM

@Limitless It's awesome!!!!!

Complex integration is great

Do you know line integrals?

Limitless

@Argon, complex integration is easier, I would hope. "easier"="more intuitive". And, no.

When I see stuff like $\int_{C}dx$, I get a little terrified.

Argon

@Limitless Easier for some stuff, harder for others. But it can be used where real analysis is hopeless

Hahaha!

@Limitless Complex contour integrals are line integrals in the complex plane

user19161

Hey @amwhy going to bed soon?

Argon

If it is over a closed region with no poles intersecting with the contour

the value of the integral depends solely on the poles in the contour.

amWhy

@WillHunting Probably. How 'bout you?

Argon

4:14 AM

@WillHunting Late for you!

user19161

@amWhy Hmm, yes. OK, see you in your dreams! =)

Limitless

@Argon !!!!. I looked at the animation explaining a line integral and had my mind blown.

Argon

@WillHunting @amWhy Good night

@Limitless It is fantastic

amWhy

@WillHunting I'll wave !

Limitless

@amWhy, Bai

user19161

4:15 AM

@Argon Are you going to appear in our dreams too?

Argon

@WillHunting If you want!

Limitless

@Argon, you'll love this guy.

user19161

@Limitless You don't love me?

Argon

@WillHunting I'm that guy from the music video, remember?

@Limitless "The board requires you to be registered and logged in to view profiles."

Limitless

@WillHunting, do I? :o

user19161

4:16 AM

@Argon Hahahaha, actually I am the guy and the girl is ... . =)

Argon

@WillHunting Does he?

@WillHunting XD

user19161

@Argon Is XD=XXX?

Limitless

@Argon, drats!!! Take a look at this.

Argon

@WillHunting $D=XX \implies XD=XXX$

$=X^3 \,\,\blacksquare$

@Limitless Cool!

user19161

@Argon QED.

Argon

4:18 AM

@WillHunting Yep

Limitless

@Argon, MY MIND IS STILL BLOWN FROM THIS GIF.

Argon

@Limitless Thats a line integral! GREAT gif!

Imagine that over $\mathbb C$

Limitless

@Argon, I like having a functional brain. I'll wait until my mind gets over $\mathbb{R}$.

Argon

@Limitless It's uncountable, so it will be a while!

Derivatives of functions can be given as complex line integrals!

@Limitless demonstrations.wolfram.com/ContourIntegralAroundASimplePole

Limitless

@Argon, darn it. Cantor just diagonaled my attempt to understand.

@Argon, CA is blowing my mind.

Hmm.

Argon

4:23 AM

@Limitless I even use it to blow the minds of my classmates who know no integration with de Moivre

I find stuff like $\sin(5 x)$

or exact forms for $\sin (\pi/5)$

Limitless

@Argon, you're a winner. I have to say, I'd adore being your classmate. I am sure we could explore so many things together.

Argon

@WillHunting !

@Limitless :)

Limitless

@Argon, I am going to update my Mathematics.tex file

Just a few minutes

Argon

What does it have?

Limitless

Basic calculus things

e.g.

$(\tan^{-1} x)'=\frac{1}{1+x^2}$

The most complex thing is probably this: $$\left(\prod_{1 \le i \le n}y_i\right)'=\sum_{1 \le i \le n}y'_i\prod_{j \ne i}y_j $$

Argon

4:27 AM

@Limitless $$\implies \int_0^\infty \frac{dx}{x^2+1} = \pi/2$$

Limitless

@Argon, why is that?

Rather, how did you intuitively know that?

Argon

@Limitless $\lim_{x \to \infty} \arctan x - \arctan 0 = \frac{\pi}{2}$

Limitless

@Argon, yes. I need more intuition for the inverse trig functions! >.<

Argon

@Limitless Well, you told me that $\arctan' x = \frac{1}{x^2+1}$!

Or, try substituting $x = \tan u$

Limitless

@Argon, yes. I knew you used the nature of integration and differentiation.

Argon

4:32 AM

@Limitless This one I find cool too, because $\frac{1}{x^2+1}$ is rational.

Limitless

$\lim_{x \to \infty}\arctan x-\arctan 0=\lim_{\tan u \to \infty}u-0?$ That looks . . . interesting.

Argon

@Limitless Be a bit careful, because $\tan$ is periodic...

Limitless

@Argon, those are the bi polar functions!

Argon

(And that is weird notation. But I'm not going to call you out)

@Limitless Hahahah

Limitless

(You are too kind.)

I should be slain for my weird notation. Slain, I say.

Argon

4:36 AM

Do you like stuff like the gamma function?

user1230219

Uh guys, completely unrelated; can anyone find an english version of this?
http://docs.lib.noaa.gov/rescue/Rarebook_treasures/QB41K431627.PDF

Argon

Good night @Limitless!

Limitless

@Argon, ah! See you around.

@Argon, and yes. Yes, I do.

user1230219

It's definitely math related, but it is a bit of a stretch I guess.

Argon

@Limitless $\Gamma(1/2) = \sqrt{\pi}$

(but you probably knew that)

Limitless

4:39 AM

@Argon, I did. :) What a lovely way to part.

Argon

Good e'en!

Limitless

@Argon, I like the "proof" that $\infty!$ is defined. :P

@user1230219, I would doubt its existence.

user19161

@anon Your comment on the question is preventing me from getting more rep. =)

user19161

@Argon I logged in again just to answer another question and get more rep.

user19161

I now have 7.5k, 2.5k more to 10k...

user19161

4:49 AM

Over and out!

amWhy

@WillHunting Yay, Will Hunting! +1 from me!

KaliMa

5:21 AM

can anyone help me with this ellipse

user19161

@amWhy Haha, I just logged in again to post an even easier method! I thought about it while pooing just now. =)

user19161

Wow, N.S. gave another two solutions.

user19161

I hope this question becomes really hot and makes me cap!

user19161

Wow N.S has added a third method. Maybe I should add a third too!

amWhy

I need 86 more rep points...

user19161

5:35 AM

@amWhy You are still here!

amWhy

@WillHunting Go for it!

@WillHunting I'm fading away...time for bed!

user19161

@amWhy Good night!

amWhy

@WillHunting See you in MY dreams!

user19161

Done! Third solution!

user19161

6:05 AM

QED.

Limitless

D'aww.

I wanted to say goodbye, @WillHunting.

user19161

6:51 AM

Ladies and gentlemen, I have capped today. Please don't upvote me anymore until the next SE day, thanks!

user19161

7:05 AM

@Limitless Bye, good night!

Limitless

@WillHunting, bai!

Night

Matt N.

7:24 AM

SNOOOOOWWW!!!

Let's hope the trains aren't delayed because of this.

Harry_Potter's_Erect_Nipples

7:59 AM

bon nuit

Transcript for

$Mathematics$ Mathematics