@PeterTamaroff I am stuck

@LittleChild Can you find $$\int_{\pi}^\pi t^2\sin ntdt$$ $$\int_{\pi}^\pi t^2\cos ntdt$$?

Yes, by parts

OK, so what's the fuzz?

but this exponential thing seems to be looping

I dont seem to be getting anywhere

:(

@PeterTamaroff That has the question I am talking of. He doesnt explain at all. The font looks like ants

@LittleChild But $e^{-inx}=\cos nx-i\sin nx$

do I have to substitute ? someone said that let $e^{-inx}$ stay as it is. It will make things easier

in teh comments to the video

@LittleChild Note $\int_a^b (f+g)=\int_a^b f+\int_a^b f$

the other 1 should be g

and both integrals have to exist

what is derivative of $cos(nx)$ ? $-sin(nx) / n$ ??

@Ethan

no its not that

$-n\sin(nx)$

ok :)

lol

$sin(\frac{nx}{t})$'s derivative would be $\frac{n}{t}.cos(\frac{nx}{t})$ right ?

$$\sum_{n\leq x} \ln(d(d(n))=\ln(2)x\ln(\ln(x))+O(x)$$

$\sum_{d\mid n} 1 = d(n)$

Please use \ for logarithms, sin and other functions..

My eyes are tired of bleeding

@LittleChild Yes, that.

cool .. beginning to see a pattern here :)

@N3buchadnezzar What are you talking about?

I was just sharing some random thoughts, browsing through some integrals on the site =)

Thanks for sharing :D

@robjohn

@PeterTamaroff yes?

@robjohn Can you help me out here? Specially with last sentence, i.e., $$\lim_{n\to\infty}f_n(n+1/2)$$

@robjohn Hello?

are all the rational curves have genus zero

do*

?

@Theorem What is a rational curve?

@PeterTamaroff : a curve which has a rational parametrization .

@Theorem And how is the genus defined?

@Theorem ?

@PeterTamaroff I'm having a look at estimating your roots $x_n$.

@AntonioVargas Thanks!

@AntonioVargas I edited the question to add something else. =) This is a lot of fun, indeed!

@AntonioVargas Are you there?

@PeterTamaroff Yessir.

@AntonioVargas Did you see what I added?

I remember $\sum_{k=0}^{n} n^k/k!$ being estimated on MSE a while ago, so I'm trying to find it.

The maxima seem to go to zero as $1/\sqrt n$.

@AntonioVargas Oh, yes.

@AntonioVargas $$e^{-n}\sum_{k=0}^n \frac{n^k}{k!}\to \frac 1 2$$

I figure some proof out there should be able to be modified to get the alternating case.

It can probably be done from the integral representation using Laplace's method.

@AntonioVargas Ah! I learned Laplace's method just some days ago.

Wouldn't your integral representation help?

I would bet so.

Greetings noble minds!

@N3buchadnezzar today my research led me to a mind-blowing result that involves an infinite sum of cosine integrals.

Wop wop

Reminds me of an old putam problem

@PeterTamaroff sorry, afk for a while

@N3buchadnezzar on the other hand there might be a very easy trick around that I simply don't see (and no mind-blowing thing around) :-)

:-)

I found a easy and cute integral today

@N3buchadnezzar Which one?

Mind you this integral is very simple, but the idea in it is just brilliant

@N3buchadnezzar Minded.

@N3buchadnezzar OK, the appearance of $2^2$ already suggests something is going on...

@N3buchadnezzar this statement is annoying to me. Really! (especially the last part with the solutions)

:-)

@Chris'swisesister Bear in mind, English is not my first language ;)

howdy

@PeterTamaroff It is just there to get rid of some pesky fractions, the clever ting is that one does not have to compute explicit formulas for $\psi$ and $\varphi$, but instead one uses that $\psi(\psi-1)=1$ and $\varphi(\varphi-1)=1$.

I thought that was quite cute blushes

@N3buchadnezzar nice. Here is the cute one I was talking about $$\sum_{k=1}^{\infty} \text{Ci}(2 \pi k)$$

Nice integral.

@Chris'swisesister Where $\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos x}{x} \,\mathrm{d}x$ right?

@N3buchadnezzar mathworld.wolfram.com/CosineIntegral.html

Pesky minus sign.

aka 'Cosine Integral'

@Chris'swisesister Maple gives $-0.3860783245$

That's corect.

@N3buchadnezzar this is actually $1/2(1/2-\gamma)$

The inverse symbol generator gave me frac(\gamma*50)+(3)

I suspected something was up :p

@Chris'swisesister Did you use the sum formula for the Ci function, listed on that site?

@N3buchadnezzar yes.

=)

=)

Did you see the LambertW integral?

@N3buchadnezzar do you mean Lambert W function? I never heard of LambertW integral.

Per say it is not defined as anything special it is just the fact that

$$ \int_0^\infty \operatorname{LambertW}\left( \frac{1}{x^2} \right) \,\mathrm{d}x = \sqrt{2\pi} $$

@PeterTamaroff your roots are too much for me right now, it seems. Szego studied some similar ones a long time ago but his paper is in German so I've never been able to get much out of it.

@N3buchadnezzar this integral also appears on wiki

Yes

@N3buchadnezzar I need to leave now. Maybe you give me some hints on the sum I posted above (next time). :-)

bye

bye

n00b question: if the statement 2/(x+2) isn't a polynomial. What is it?

@Nick It's called a rational function.

@AntonioVargas: Thank you, that explains a lot. But I still dont understand why sqrt(x) isn't a polynomial.

@Nick A polynomial only has integer powers. That has a one-half power, $\sqrt{x} = x^{1/2}$, so it isn't a polynomial.

@AntonioVargas: Yes, I know that but why is that? The technical definition of a polynomial includes the statement "A polynomial can have constants, variables and exponents, but never division by a variable.". In x^{1/2}$ , Where is the division by a variable.

@Nick You're right, that's not division. And that's also not the whole definition of a polynomial.

From wikipedia: "A polynomial is either zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a constant (called the coefficient of the term) and a finite number of variables (usually represented by letters), also called indeterminates, raised to whole number powers."

whole number powers... that was the magic word

aww, thank you

np

@N3buchadnezzar Thanks, that was fun.

@AntonioVargas The other one is interesting too =)