Mathematics - 2014-07-13 (page 1 of 2)

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robjohn

00:13

@Chris'ssis Which relation? The one I mentioned on chat this morning, or the one in an earlier answer?

@skullpatrol What kind of fun will be had?

skullpatrol

@robjohn just commenting on the game as it happens...

robjohn

@skullpatrol Ah, the soccer game?

skullpatrol

@robjohn Yes the world cup final: Argentina vs Germany

robjohn

@skullpatrol Yes, I saw at lunch today that it was between Argentina and Germany.

skullpatrol

@robjohn This is the third time these two teams have met in the final, each has won once...

skullpatrol

00:33

All square at 1-1, Germany over Argentina in Rome, and Argentina over Germany in Mexico
both of those were back in the Maradona days, the 90 and 86 cups respectively

The big question now is will we have the Messi days,
or will the refs steal them away like they did in the last WC?

2 hours later…

Ben

03:04

the first term of a sequence is one. the second term is two. the third term is 3. the 4th term is 5. the 5th term is 7. all terms after the second term repeat and follow the order 3,5,7; for example, the first 11 numbers are 1,2,3,5,7,3,5,7,3,5,7...

What is the 100th and 101th term of the sequence

I thought of using a1+(n-1)d because the every multiple of 3's term is 3...

Rajesh D

03:39

@Rob : I am a La Albiceleste fan!

(Argentina!)

r9m

03:51

Argentina :D .. YYeeaaaHH !! ...

Sawarnik

04:15

@r9m :P

@r9m here? :<

Vrouvrou

04:33

hello

can someone help me please{math.stackexchange.com/questions/865702/density-and-convergence}

r9m

04:48

@Sawarnik y u ping me so many times yesterday ? :P

revenge ping @Sawarnik

1 hour later…

Sawarnik

06:06

@r9m because u r never here!

now?

r9m

@Sawarnik ya .. Hi :D

r9m

06:39

oh well .. now the Imp is not here :P

Chris's sis

07:17

Greetings

:16557385

user image

@robjohn Do you have a proof for that?

robjohn

@Chris'ssis let me look at it...

@Chris'ssis Yes... it is pretty simple.

Chris's sis

I guessed there must be such a relation, but I didn't see it when I needed it.

robjohn

@Chris'ssis It is pretty much just looking at the sum of $\frac1{k^an^b}$... The other two terms are the ones where the number to the power $a$ is greater or the number to the power $b$ is greater.

@Chris'ssis what is left is the terms where those numbers are equal

Chris's sis

@robjohn Indeed. Did you use this formula before?

robjohn

@Chris'ssis care needs to be taken when $a=1$ or $b=1$ because the convergence is not absolute immediately

Chris's sis

07:29

I see.

Chris's sis

07:53

@robjohn you're silent sometimes, but when you strike again,you do it deadly ... :-))))))))

That proof is a masterpiece. And taking into account that these series are pretty hard ...

robjohn

@Chris'ssis There are just a few tricks that cover a good deal of the cases... That one I mentioned has been quite useful

@Chris'ssis Then of course that one with the Euler series :-)

Chris's sis

@robjohn Yeah, that is well-known. I often use it.

@robjohn These identities like the one you used are pretty interesting. I knew they must exist, but I didn't know them. I'm not referring to the Euler series, but to the identity in the picture I posted.

user image

robjohn

@Chris'ssis That is just breaking the sum into a few parts depending on the relative size of the terms. Ive used it in many of these evaluations. I thought for sure that they were common.

Chris's sis

@robjohn Those sums are known as Euler-Zagier sums (excepting the sum that is made of the first term)

robjohn

@Chris'ssis well, the first term is just $\zeta(a)\zeta(b)$

Chris's sis

08:03

@robjohn Yeap. By the way, that identity looks much better if written in zeta :D

robjohn

@Chris'ssis It often turns out that we can use $\sum\limits_n\frac{H_n^{(a)}}{n^b}+\sum\limits_n\frac{H_n^{(b)}}{n^a} =\zeta(a)\zeta(b)+\zeta(a+b)$

Chris's sis

@robjohn summation by parts.

robjohn

@Chris'ssis that is just the identity in the image you posted

Chris's sis

@robjohn Indeed.

robjohn

The image identity is actually, $$\sum\limits_n\frac{H_{n-1}^{(a)}}{n^b}+\sum\limits_n\frac{H_{n-1}^{(b)}}{n^a} =\zeta(a)\zeta(b)-\zeta(a+b)$$ but it is easy to get from one to the other.

Chris's sis

08:17

@robjohn Now, if I combine your work with my work, I get an elementary evaluation of $$\sum_{n=1}^{\infty} \frac{1}{n^4}\left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)$$

:D

robjohn

@Chris'ssis yep... That is the same sum :-)

Chris's sis

These series are really really nice ...

robjohn

@Chris'ssis That was the last one on the image you posted the other day. Did you have another method to evaluate it?

Chris's sis

@robjohn I initially thought I can do it elementarily, but I was deceived by a formula ...

@robjohn I used this paper to finish my work there projecteuclid.org/download/pdf_1/euclid.em/1047674270. See the page 23.

robjohn

@Chris'ssis the multiple Euler series results might give some aid in the 3 variable sum you were looking at

Chris's sis

08:24

@robjohn One thing is clear: the best way to solve these problems is to bring all in the area of series. Working with integrals is proving to be a very hard task (for hard problems).

@robjohn Just look at the integral representation of the series we worked with and see what's there ... :-)

The hell

:-)

robjohn

@Chris'ssis That has been my experience, too.

Chris's sis

@robjohn Yeah, it might help. That seems to be the kind of a very hard nut.

robjohn

@Chris'ssis is that in that paper?

Chris's sis

@robjohn No. I don't even know if that series is evaluated in any paper (the 3 variable version I mean).

robjohn

@Chris'ssis I was talking about...

5 mins ago, by Chris's sis

@robjohn Just look at the integral representation of the series we worked with and see what's there ... :-)

Chris's sis

08:34

@robjohn see this one, page $5$ arxiv.org/pdf/math/0406401.pdf

robjohn

On page 21 they talk about the fact I mentioned yesterday about the sum of the exponents being odd...

Chris's sis

@robjohn Yeah, but this is a known thing that also appears on mathworld page and many other papers.

There are some problems when we deal with the even cases.

brb

robjohn

@Chris'ssis yeah, but the Euler sum I posted a while back (that appears in the first paper you linked) for $$\sum_{n=1}^\infty\frac{H_n}{n^q}$$ works for all $q\ge2$

Chris's sis

@robjohn I wonder why @r9m is so silent these days ...

@r9m don't you have some nice ideas to share on that series I posted above? :-)))

bbl

r9m

09:11

@Chris'ssis oh well ... NOISE :|

@Chris'ssis no-idea :-)

r9m

10:05

@Sawarnik Hi

Sawarnik

10:23

@r9m yay! hi!

did you do the geometry i gave?

r9m

@Sawarnik 'sup ? :D

@Sawarnik nah

Sawarnik

@r9m Do it ... its very very easy :P

@r9m Have you ever watched cricket?

r9m

@Sawarnik ya I know .. thats why I didn't do it :P

@Sawarnik nope .. I don't watch cricket

Sawarnik

@r9m :'(

But still can you a believe a 100 and then a 200 run stand in the last wicket in the same match!

@r9m :O Is there a perfect weather for sleeping?

r9m

@Sawarnik yes .. it rained here :P ... the weather was ideal for sleeping :P

Sawarnik

10:29

Oh.

1 hour later…

Lost1

11:44

math.stackexchange.com/questions/493739/… Iroinically, this question was answered by someone called Doc...

Balarka Sen

@Lost1 $x = y = z$..?

sammyg

In a strict mathematical notation, is it illegal to start a new line with the equal sign?

"As you would continue a sentance in English by
returning to the next line?"

Balarka Sen

Oh, distinct positive real. Right.

That's actually a good question.

Lost1

^@BalarkaSen I was merely pointing out the question was 'i had a bad cough' and the answer was given by someone called 'Doc'

for a moment, i thought this guy did it as a joke haha

Balarka Sen

@Lost1 Oh right

r9m

11:54

@Lost1 lol XD

Balarka Sen

@r9m I have commutative algebra class tomorrow.

I am positively scared by what's coming in there.

r9m

@BalarkaSen OMG ,, where ? :D

Balarka Sen

@r9m Oh, you know, Belur.

Lost1

isn't term over in most places? where are you from?

Balarka Sen

@Lost1 India.

r9m

11:57

@BalarkaSen Ic ic ,. I haven't read comm alg yet :D .. must be interesting :D

Balarka Sen

@r9m They say it is. But they also say that they are scary.

Sawarnik

@shadow10 Why is everyone Bengalis here?

Balarka Sen

12:13

@Sawarnik math.stackexchange.com/questions/613688/….

Fun, isn't it?

Sawarnik

12:30

@BalarkaSen Oh, I have seen it :)

Mats Granvik

What is the main trouble in trying to solve $1+\frac{1}{2^x}-\frac{2}{3^x}=0$ ?

r9m

@MatsGranvik real solutions ? or solutions in $\mathbb{C}$

Mats Granvik

Solutions in complex numbers. I don't know if this particular one equation has any other solutions.

Except x = 0

r9m

@MatsGranvik put it in W|A :D

Mats Granvik

12:45

@r9m Right. I did that now. According to wolfram alpha the only solution is x=0.

r9m

@MatsGranvik wolframalpha.com/input/…

Mats Granvik

@r9m I see.

3 hours later…

china math

16:14

hello,everyone

r9m

@chinamath Hi :D

china math

HaHa,hello,r9m

@r9m

r9m

@chinamath I'm seeing you for the first time in this chat room :-)

china math

yes, this is first time in this chat room

and my English is poor,sorry

What is this place, because the first time I come,sorry,I don't know

and @r9m,I have hard math problem can you see it,

my frends give me,and Now I can't solve it,

r9m

@chinamath sure .. I can atleast take a look and try :P

china math

16:20

HaHa,Thank you ,this is a function equation

Find all the function $f$ ,such $f$ have Twice differentiable on $R$, and for $xy\neq 0$, then have

\[xf\left(\dfrac{f(y)}{x}\right)=yf\left(\dfrac{f(x)}{y}\right)\]

my idea is Differentiate the equation for x

and I found is f(f(y)/x))-f(y)/x*f'(f(y)/x))=f'(x)/y*f(f(x)/y))

so I want let f(y)/x=u.then I can't.

Now,it is clear that have found that f (x) = 0 is such it,but I can't sure have other function such it

r9m

okay .. I'll try to think about it :-)

china math

Thank you.

r9m

are you from China ? :D

china math

and my Email is [email protected], if you have solution,can you send my email.Thank you

yes

r9m

okay :-)

china math

16:29

and are you from which conutry?Thank you

r9m

@chinamath India :) ..

china math

oh,Thank you

r9m

@chinamath You post some really nice problems ! :D ... are you a college/undergraduate student ?

china math

yes,I'm undergraduate student. some problem is from frends.

r9m

@chinamath Which city are you from ? ( .. if you don't mind me asking :) )

china math

16:34

oh,you know china city? china have many city.

r9m

@chinamath ya ... there are so many cities in China ... its a vast nation :) ... which city are you from ? :-)

china math

I was born Jiangxi province,can you hear it?

r9m

@chinamath ya .. I've heard about it :-) ... I watch too many Chinese movies :P

china math

ya.haha Welcome to China to play

r9m

@chinamath can you tell me about the source of this problem math.stackexchange.com/questions/718166/… ? :-)

china math

16:47

no,this problem is not from sackexchange,,is my frends ask me,I don't know this problem is from where, maybe Which mathematical olympiad

r9m

I see ... I tried to do that problem .. but found no simple solution X_x ..

its a nice problem :-)

@chinamath do you have a simpler solution that the ones already posted ? :)

china math

Now,I have no solution.Thank you

r9m

okay .. thanks :)

Vibhav Pant

17:19

hello

@r9m I need your help

I started with Bartle and Sherbert (real analysis) recently, and I'm on Limits

Pedro Tamaroff

@VibhavPant Maybe I can help you.

Daniel Fischer

Hola @Pedro.

Pedro Tamaroff

Hello Daniel. How's it going?

S**t is going down today at 5 p.m.

=D

Daniel Fischer

Fine. After almost four and a half months, finally Ross' answer got its eleventh upvote, and I my populist badge ;)

Vibhav Pant

@Pedro This chapter makes extensive use of the $\epsilon$-$\delta$ definition of limits, in order to prove simple results

Pedro Tamaroff

17:25

@VibhavPant OK.

@DanielFischer Ah. Badges!

It's been a while since I got one.

Vibhav Pant

@Pedro Do I still need to read them?

Pedro Tamaroff

The $\varepsilon$-$\delta$ definition should be something you know like a poem, as Spivak says.

Your comment strikes me as "why is he using something complicated to prove something simple"?

You just have to get used to the $\varepsilon$-$\delta$ definition.

Vibhav Pant

Yep

Pedro Tamaroff

And getting used to it means reading it many times, doing many exercises, proving many limits equal something using it, &c.

@DanielFischer I finished all my midterms. Now I can hibernate. =D

skullpatrol

are you gonna watch the game?

Daniel Fischer

17:28

@PedroTamaroff I thought you'd go to the north, it's summer there, you can't hibernate in summer, can you?

r9m

GAME ON !!!!! .. I've been painting my face :D HOHO !!!

Pedro Tamaroff

@DanielFischer True. I'll hibernate just a few days, then leave.

@r9m You're German?

>:)

skullpatrol

1 hour 30 min

r9m

@PedroTamaroff Argentina :D

robjohn

@DanielFischer where is Pedro?

Pedro Tamaroff

17:30

@r9m You're Argentinean, or just cheering for Argentina?

Daniel Fischer

@robjohn At the moment, somewhere in Argentina, as far as I know.

robjohn

@DanielFischer so it's winter there...

Daniel Fischer

But he plans to visit Ted.

china math

hello,@robjohn

r9m

@PedroTamaroff of course cheering :P ... I'm Indian :P

Pedro Tamaroff

17:31

@r9m Ah! I did see a video where many of you were cheering. I was impressed.

robjohn

@DanielFischer Oh, I get the comment... You'd think he'd come to this hemisphere :-)

china math

I know @robjohn,it's severe

robjohn

@chinamath hey there

china math

Haha

Daniel Fischer

@robjohn If he visits Ted, he does. Georgia is only relatively south. Viewed from Argentina, it is far north.

china math

17:32

e

r9m

@VibhavPant Hai :D

skullpatrol

Good luck to both of your teams @DanielFischer @PedroTamaroff may the better team win, without any interference from the "officials." :-)

Daniel Fischer

@skullpatrol Cristina Kirchner > Angela Merkel $\implies$ Argentina wins.

china math

Hello,@DanielFischer,@robiohn and other ,Now I have function equation problem,can you see it?

Vibhav Pant

@r9m this book is pretty dense

Pedro Tamaroff

17:36

@DanielFischer CFK > Merkel? WAT.

Daniel Fischer

@PedroTamaroff Well, for one she has the advantage of being far away from where I live ;)

china math

Find all the function $f$ ,such $f$ have Twice differentiable on $R$, and for $xy\neq 0$, then have
xf(f(y)/x)=yf(f(x)/y)

Pedro Tamaroff

@DanielFischer Hehehhe.

r9m

@VibhavPant 'dense' ?! :P ..

china math

Find all the function $f$ ,such $f$ have Twice differentiable on $R$, and for $xy\neq 0$, then have
xf(f(y)/x)=yf(f(x)/y)

this problem is hard,and is my frends ask me

Vibhav Pant

17:38

@r9m yeah. I'm still new to the $\epsilon$-$\delta$ definition, and most examples use this one

N3buchadnezzar

Hmm

$$ \int_{1/4}^{1/2} \lfloor \log \lfloor 1/x \rfloor \rfloor \,\mathrm{d}x$$

r9m

@VibhavPant ya ... its what Pedro said .. $\epsilon-\delta$ proofs are the core of real-analysis :) .. getting used to it is good for health :D

china math

this integral is not hard

you can find the (1/4.1/3),and (1/3.1/2)

Vibhav Pant

alright then

china math

when 1/4<x<1/3,then [1/x]=3,

and 1/3<x<1/2,then [x]=2

[1/x]=2

r9m

17:45

@robjohn I have one integral inequality problem :D

robjohn

@r9m which is?

N3buchadnezzar

@chinamath So the integral equals $1/12$ thanks =)

china math

No thank ,

hahah

r9m

@robjohn This one :D

skullpatrol

@DanielFischer ;-)

N3buchadnezzar

17:49

\begin{align*}
\int_{1/4}^{1/2} \lfloor \log \lfloor 1/x \rfloor \rfloor \,\mathrm{d}x
& = \int_{1/4}^{1/3} \lfloor \log \lfloor 1/x \rfloor \rfloor \,\mathrm{d}x
+ \int_{1/3}^{1/2} \lfloor \log \lfloor 1/x \rfloor \rfloor \,\mathrm{d}x \\
& = \int_{1/4}^{1/3} \lfloor \log 3 \rfloor \,\mathrm{d}x
+ \int_{1/3}^{1/2} \lfloor \log 2 \rfloor \rfloor \,\mathrm{d}x \\
& = \int_{1/4}^{1/3} \lfloor \log 3 \rfloor \,\mathrm{d}x \\
& = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}
\end{align*}

Daniel Fischer

@skullpatrol Ever heard of Lionel Messi?

N3buchadnezzar

@DanielFischer David vs Goliat

r9m

@N3buchadnezzar XD .. truth is spoken :D

skullpatrol

The refs can swing the game too

Daniel Fischer

@N3buchadnezzar Poor Goliath never had a chance.

china math

17:52

@r9m,I have see your problem, maybe can use Holder inequality?

Ted Shifrin

ah, I forgot, it's gonna be @DanielF versus @Pedro in here

N3buchadnezzar

@DanielFischer Messi is really good and Argentina does have a good chance. But in the last three games Messi has always been marked by atleast 3 players.. He is the player who has run the least (after fred). The reason Argentina is in the final is not because of Messi, but because Argentina has only let in 3 goals.

Daniel Fischer

@TedShifrin Not really. We might argue whether Cristina Kirchner or Angela Merkel sucks more, but as far as the football is concerned, we both know that Messi is the best player on the pitch. By far.

Pedro Tamaroff

@TedShifrin I've been told taxis from Newark to NJ aren't that expensive.

Ted Shifrin

mr @Pedro is here!

well, every day you wait, the prices go up, @Pedro ... or availability goes down at lower prices

N3buchadnezzar

17:56

Germany lacks a superstar, but in return their team has a whole is more tied tohgeter and not dependent on one particular player. Although the loss of either Müller or Schweinsteiger could prove fatal.

Alizter

Hmm the bookies are offering £100 if Argentina win. They don't seem too positive about Argentina winning.

Daniel Fischer

@Alizter They weren't positive about Uruguay winning either, in 1950.

Ted Shifrin

And @DanielF was there to see it :)

Daniel Fischer

@TedShifrin Not quite.

N3buchadnezzar

Why does Ted come with old man jokes?

r9m

17:59

@chinamath $\sqrt{x}$ is trouble, I can't eliminate it with plain Holder :|

Daniel Fischer

But I'm neither there to see Argentina or Germany win today.

Ted Shifrin

Because I'm entitled to, @N3 :P

Pedro Tamaroff

@TedShifrin I know, I know...

Ted Shifrin

Just like, in principle, Jews are allowed to tell Jewish jokes and gays are allowed to tell gay jokes.

So I can do all three :P

AWertheim

Lol, in principle :)

N3buchadnezzar

18:00

It is like yoda mocking Darth Sidious for being old.

Ted Shifrin

heya @AWertheim!!

AWertheim

Hello @TedShifrin :)

china math

@r9m,Oh.

Vibhav Pant

Someone tell me if I am right: A cluster point $c$ for a set $A$ exists if for at least one $x\in A$, the difference between $x$ and $c$ can be made as small as possible.

Alizter

If Germany win then the European economy will benefit because of celebration and happiness. If Argentina win the European economy will benefit because of sad Germans drinking a lot of beer.

AWertheim

18:02

I was terribly disappointed to find that this movie (imdb.com/title/tt2333804) has absolutely nothing to do with Hilbert's Nullstellensatz...

Ted Shifrin

@Vibhav: Obviously a given $x$ has a certain distance from $c$. You need to say it more carefully.

N3buchadnezzar

Such football, much balls, wow.

Alizter

@AWertheim Doesn't he 'prove' that the universe is equal to 0?

china math

hello,everyone, can you see this problem Find all the function $f$ ,such $f$ have Twice differentiable on $R$, and for $xy\neq 0$, then have
xf(f(y)/x)=yf(f(x)/y)

AWertheim

Lol @Alizter, not sure. It hasn't come out in the US yet. I was just making a stupid joke :)

N3buchadnezzar

18:03

@Alizter Germans beer consumptions is independant of the World cup. They constantly drink anyways.

Pedro Tamaroff

@VibhavPant No. Cluster point means that there are points of $A$ different from $c$ arbitrarily close to $c$.

AWertheim

It looks like any math in there is going to be gibberish though, so there's that =P

Alizter

@N3buchadnezzar Think of a normal distribution around the world cup however translated upwards.

r9m

@chinamath see the right panel ,.. I have starred your problem ... if anyone replies to that you will get notified in your inbox :)

china math

oh,Thank you,my email is [email protected]

Alizter

18:05

In Germany they have no speed limits on large roads and you can drink and buy from 16. Yet less traffic accidents than Britain even with the population difference.

And this one time, don't tell anybody, I J-walked. Tee hee.

Vibhav Pant

@Pedro Im taking $c$ as the cluster point

N3buchadnezzar

@Alizter ^^

@Alizter Got any fun small integration problems?

AWertheim

Hmmm. Does anyone here read Japanese well?

china math

I found stackexchange @Jack D'Aurizio is very good

Alizter

@N3buchadnezzar Yes actually

china math

18:08

very very good math

someone agree it?

Alizter

user image

Pedro Tamaroff

@VibhavPant Aha.

N3buchadnezzar

@chinamath Much math, such solve, many answers

Alizter

@N3buchadnezzar

r9m

@chinamath yes .. he posts really nice solutions :D

N3buchadnezzar

18:09

@Alizter Complete Elliptic integral?

china math

@N3buchadnezzar,yes

Pedro Tamaroff

A cluster point of a set $A$ (in my books) is defined as a point such that for any $\varepsilon>0$, there is $a\in A$ with $|c-a|<\varepsilon$.

Alizter

@N3buchadnezzar yeah

Pedro Tamaroff

But nomenclature varies.

Vibhav Pant

same here

N3buchadnezzar

18:13

Veni, veni, veni

china math

I think this integarl can Exchange order

\int_{0}^{1}\int_{0}^{\pi/2}\left(\dfrac{\sqrt{1-k^2\sin^2{\theta}}-1}{k}\right) d\theta dk

Alizter

$\displaystyle\int_{0}^{1}\int_{0}^{\pi/2}\left(\dfrac{\sqrt{1-k^2\sin^2{\theta}‌}-1}{k}\right) d\theta dk$

N3buchadnezzar

@Alizter Switching the order seems tough

Alizter

@N3buchadnezzar It's not my problem. It was another persons creation. It is definitely solvable however way beyond my capabilities. You are good with integrals so I thought why not show you.

N3buchadnezzar

@chinamath I think it should be $-\pi/2$ and not $-1$

china math

18:20

@N3buchadnezzar,I use this Instead $\int_{0}^{\pi/2}1d\theta=\pi/2.$

N3buchadnezzar

@chinamath I agree

china math

wolframalpha.com/input/…

Vibhav Pant

Every point in $\mathbb{R}$ is a cluster point of $\mathbb{R}$, right?

r9m

ya

Balarka Sen

@N3buchadnezzar haha

Vibhav Pant

18:28

Guess I'm understanding this stuff then :D

Balarka Sen

What stuff, @VibhavPant?

I thought you were doing NT

N3buchadnezzar

@chinamath Any idea what to do next?

Vibhav Pant

@BalarkaSen CMI's exam is right around the corner, so I am somewhat rushing with the syllabus

I gave half a month to NT (which was mostly divisibility)

china math

@N3buchadnezzar,But I can't works next,sorry

Balarka Sen

@VibhavPant tsk tsk Should've read a decent text on NT.

r9m

18:30

@BalarkaSen Vibhav is a good boy ... he's learning real-analysis :D ... unlike you .. you are a mathematical outlaw ... rogue mathematician :P

Balarka Sen

Divisibility is totally not everything even in elementary number theory.

N3buchadnezzar

@chinamath I tried parts but no, i tried switching but no

Balarka Sen

@r9m STAHP. I hate real analysis

Vibhav Pant

D:

Pedro Tamaroff

@BalarkaSen tsk tsk tsk

Balarka Sen

18:31

On the other hand, complex analysis is my friend.

Pedro Tamaroff

You cannot hate something you don't know about.

N3buchadnezzar

@BalarkaSen Do you *real*y hate it?

Balarka Sen

@PedroTamaroff Comm. Alg. classes tomorrow.

@PedroTamaroff I do several things that I shouldn't or I can't.

I admit that I never tried to siriusly study real analysis.

china math

user image

Balarka Sen

@chinamath Yes.

china math

18:33

maybe use this

E(k)-pi/2/k

Balarka Sen

Good idea. But then we can't.

china math

is series
$$-\sum_{n=1}^{\infty}\dfrac{1}{2n(2n-1)}\left(\dfrac{(2n-1)!!}{(2n)!!}\right)^2=RHS$$

so we must find this seris sum?

N3buchadnezzar

@chinamath I think I have an idea

china math

and it is know that $\int_0}^{\pi/2}\ln{(\cos{x/2})}dx=G-\pi/2\cdot\ln{2}$

N3buchadnezzar

$$ \frac{\mathrm{d}E(k)}{\mathrm{d}k} = \frac{E(k) - K(k)}{k}$$

china math

18:37

then?

N3buchadnezzar

Solve for $E(k)/k$, then replace the expression

china math

But K(k) is very ugly.

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form : f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to th...

N3buchadnezzar

Hmm, right

china math

18:51

user image

sorry,some wrong.

user image

follow I think we must find this sum?

maybe can use (arcsinx)^2

N3buchadnezzar

Doubt it

Chris's sis

@robjohn I just discovered a very powerful identity that will help me compute a famous series in a very easy way.

china math

@Chris'ssis,can you say it? Thank you

Chris's sis

@chinamath You'll find it in my book (unpublished yet).

N3buchadnezzar

@Chris'ssis Did you post/see this one?

Chris's sis

18:58

@N3buchadnezzar No.

china math

oh,When published
? Thank you

N3buchadnezzar

How can we prove that $$ \int_0^2 \sqrt{x + \sqrt{ x + \sqrt{ x + \cdots\,}\,}\,} \,\mathrm{d}x $$ converges? Finding the value 19/6 is very easy, but I have some problems proving convergence.

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