@robjohn Yeah, you can't also be circling around the origin at the same time. But the fact that a rotation of $4 \pi$ brings you back to where you started is consistent with the branch point behaving like a square-root singularity. I was looking for that consistency, but I wasn't quite seeing it.
There must be a post on meta.stackexchange somewhere about why they allow someone to vote to delete a question again (after it is undeleted) - but not vote to close it again, after it is reopened. But I can't seem to find a post on meta.SE about it. Does anyone happen to have a link?
@RandomVariable if you look at the branch cut for $z\lt-\frac1e$, it is a crossing of two sheets, $\mathrm{W}_{-1}$ branching to $\mathrm{W}_0$ which passes through $\mathrm{W}_0$ branching to $\mathrm{W}_1$. For $-\frac1e\lt z\lt0$, $\mathrm{W}_{-1}$ branches to $\mathrm{W}_1$.
@CarlMummert Why do you think that people that are very good at integrals, series and limit never vote for closing those questions that you usually vote for closing?
@CarlMummert Maybe they understand the beauty and the potential of the questions and the answers from which they can learn something new and become better.
@CarlMummert as a professor, if you always attend only problems from the textbooks I doubt you'll ever be great.
I don't like the mediocrity at all, that's why I put so much price on these questions. I cannot imagine myself without attending problems coming from my research or others research.
@CarlMummert some people have great dreams, some dream to become like Ramanujan or even more in the area of integrals, series and limits - I'm such an example. I need what you usually like to vote for closing.
@MikeMiller If you're not busy, can you tell me a bit about Hatcher's discussion on cycles and manifolds on page 109? I have only just opened the book to read it carefully once again without skipping minute details before finishing the exercises, and this paragraph attracted my attention.
Can you do it where it's not a manifold at the $n-3$ simplex itself, rather than on the interior? (If you've got a 3-dim simplex, say, the interior is a silly thing. I'd like to see it fail here.)
@RandomVariable in fact, they discuss the image of the curve $-1/e+\epsilon e^{i t}$ for $-\pi < t\leq \pi$ on page 18 of that paper. so you might be able to extract an answer from there.
I have a doubt regarding the evaluation of the following integral :
$$
\int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)}
{(1 + 3x^2)\sqrt{1 + x^2}}\,du = \frac{\pi^2\sqrt{2}}{60}.
$$
Could anybody please help by offering useful hints or solutions? I think very difficul...
there's a difference between the interest of the mathematical problem being posed, and the quality of the question itself as formulated by the proposer
i'm not sure where i come down on what standards should be applied, but there does need to be a balance
i have sat on several classes with the grads, and most of the time they just snore and pretend to be listening while fiddling with their phones. disgusting.
@Semiclassical That paper was actually the original source of my confusion. I didn't understand how three branches could share a square-root singularity. Robjohn posted an answer to my question, and that was what I was asking him about.
one of them told the head of the department that he likes to differentiate while trying to get admitted. and he was supposed to be doing lots of more stuff than just differentiating in his undegrads. good lord, indian educational system sucks so much.
unfortunately, in the US we've watered down our education so that we have students getting through an undergraduate math degree who can't do much of anything ...
The closed form of the integral
$$\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$
I have a doubt regarding the evaluation of the following integral :
$$
\int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)}
{(1 + 3x^2)\sqrt{1 + x^2}}\,du = \frac{\pi^2\sqrt{2}}{60}.
$$
Could anybody please help by offering useful hints or solutions? I think very difficul...
