If you let $w=\frac{z+1}{z-1}$, then $dw=\frac{-2dz}{(z-1)^2}$, and $(w-1)^2(z-1)^2=4$, so you end up with an integral $$-2\int_{\gamma}\frac{\log w}{(w-1)^2}dz$$
Here $\gamma$ is a circle that contains $1$ oriented clockwise.
By Cauchy, the integral equals $2\cdot 2\pi i (\log w)'_{w=1}=4\pi i$.
I am wondering why a period integral with rational-function integrand can always be written with a polynomial integrand by "introducting more variables"
I don't know any of the algebraic geometry necessary to understand at that level though
there is literally a proof method called just-do-it
as expected, you just-do-it
@evinda another useful thing for verifying the most basic of facts: unpackage definitions. What does it mean for $I\cap J$ to be an ideal? Well it means [blah] and [blah]. Just prove both [blah]s are true, using the hypotheses that $I$ and $J$ are ideals!
even the primitive roots of unity (of which there are phi(n), same as the degree of E|K) are not a basis, as the sum of them is an integer (specifically a coefficient of a cyclotomic polynomial, even more specifically equal to mu(n)) (which also means they cannot span the whole space either)
the only inseparable extensions exist in positive characteristic
@skullpatrol he's on the shortlist if rob or I ever feel like stepping down or expanding. I suppose I don't see any argument against expanding, either.
Right. I'm asking about non-simply connected surfaces, like the punctured plane with the metric it gets by sitting in the plane. It seems to me that the holonomy of this must be zero even though it's not simply connected.
Ok, thanks, that's what I was confused about - your wording made me think you were saying "the holonomy around any nontrivial curve is nonzero". Thanks for bearing with me!
You're an ant living in a surface, @Pedro, and you move a tangent vector along a curve, turning it as you go so that its derivative is only normal to the surface (so to the ant it is moving parallel, i.e., not turning)
@Ted I understand your example. I just thought when I read it you were saying something about non-null curves on anything.
Oh, I suppose it can be done lots of ways in terms of how the ant moves. But if you only consider where the tangent vectors end up, rather than the path they take, it's unique.
What do I tell them? I feel like given that I've turned in crappy problem sets and done badly on exams, they probably think I'm a terrible student and won't want to talk to me ...
That's not how professors work. :P Trust me, they're more than glad to talk to you, and they have your best interests in mind if you talk to them about how you're doing in the class, what you should do going forwards, whether you should keep taking or not, etc.
Sorry to bug you guys again, but we have thanksgiving break starting from next week, so my professors won't have classes or office hours during which I can meet them ... should I email them to ask if I can meet them tomorrow (last day before break)? Also should I mention it's because I'm considering dropping the courses?
@M.N.C.E. (sorry for the late reply .. I was afk) .. I wanted to ask you .. Asia is a very big place .. where are you from ? :-) (If you don't mind me asking)
@M.N.C.E. oh ! I've never seen a high-school student solve such incredible integrals like that !! .. I mean I've never seen anyone from high school so motivated with special integrals ..
@r9m I browsed around MSE a few months ago and saw a couple of difficult integrals along with their solutions. I guess that was how I start to take interest in solving integrals. There isn't much to it actually... :/
The question was a PSQ (I am still not convinced that they are all off topic, especially when the question is a good one), but it had many upvoted answers. Its votes were +3/-3, so the people who wanted it gone had enough downvotes to make it to 0 and then delete.
@r9m Nah... I could undelete it, but only people with 10K or more can see it and vote to undelete.
@Pedro there's just one issue, how do you show that there is morphism by the set of cosets? Moreso, if that morphism is between $G/H$ and $S_5$, than we already assume $G/H$ is a group which is true iff $H$ is normal
Also, I don't see how to show that $S_5$ doesn't have element of order 15 anymore.. but that's solveable
What I am suggesting here is a form of self-government. Since, the system records and graphs the activity of all the users in the room, why not choose the room owner based on that data, rather than the current system? If you want to be a room owner it is only fair that you should actively participate in that room.
@robjohn Are you aware of an application of Holder's inequality which allows $\int |f|^{\frac{1}{p}'}g$ can be stated to be $\leq \int |f|^{\frac{1}{p'}}|g|^{\frac{1}{p}}$, where $p'$ is the conjugate of $p$?
@MikeMiller I got a naive question. I learnt what groups are, I learnt what dihedral groups, symmetric groups, permutation groups and matrix groups are, abelian, non-abelian etc and that they're abstract. Though I haven't seen anything proven with group theory, I only see a handful of definitions. All I see is more abstract restatements of definitions... What can we do with group theory? What's a classic example?
