you could take $f(x)\in I\implies f(x^n)\in I$ as a criterion unto itself, but it'd be nice to have a more familiar way to present that. (there might not be one)
@RandomVariable Thank you. Well, I don't have other better option, I need to highly believe in myself and work crazy hard, this is the only way to success in my opinion. I wanna created amazing questions, I wanna obtain amazing proofs, I wanna be like Ramanujan, it's one of my dreams.
i have a fairly wide vocabulary, which also means that i occasionally remember words entirely wrong. harder to keep precise definitions in your head when you have too many of them
there is something still wrong. What you have is called the cosine integral, it does not have an elementary antiderivative, hence you cannot use FTC on this.
Before you answer this OP, please read all the terms and conditions below. Thank you...
Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for fun guys. (>‿◠)✌
Before we start the contest, here are the rules of my little contest that you ...
In 1638 Galileo published Two New Sciences, in which he described his inclined plane experiment.
He discovered that the acceleration of gravity was uniform, and could be modeled mathematically by the simple equation < Distance = c * t² >.
Question: Was this the first discovery of the fact ...
I am on mobile since my brother took my pc along with his new one because he didn't have the time to spot and copy/reinstall all programs/projects he had on mine
Here is some progress towards a solution. To review, we have $\psi:\mathbb{R}\to\mathbb{R}^{3\times 3}$ such that $\psi_0=0$. (Notation: I take $\psi_0^{(n)}:=\frac{d^n\psi}{ds^n}\bigr|_0$). We further assume that $\psi_0'$ is a special orthogonal matrix (determinant 1) and $\psi''(s)=(A+Bs)\psi'...
@Semiclassical: I could access it at uni, strangely enough it doesn't work at home. Can you tell me which was reference number 23 in this paper? (scitation.aip.org/content/aip/journal/jmp/8/4/10.1063/1.1705306) I think it was something from Snider, but I don't recall the title.
@DanielFischer With regard to an answer I saw posted on MSE, does $f(z) = z^{z}(1-z)^{1-z}$ not have a branch point at infinity? It certainly appears that $f(1/z)$ doesn't have a branch point at the origin, but I feel like I'm doing something careless.
@RandomVariable For $\lvert z\rvert > 1$, we can write that as $$z\cdot z^{z-1}(1-z)^{1-z} = z\left(1-\frac{1}{z}\right)^{1-z}$$ using the principal branch of the logarithm for $(1-z^{-1})^{1-z} = \exp\left((1-z)\log (1-z^{-1})\right)$ (that is one branch of the function defined on $\lvert z\rvert > 1$), so $\infty$ can be an isolated singularity, hence is not a branch point.
it also comes up in the classical mechanics context of hamiltonian evolution
which is perhaps not so surprising when one considers how many analogies there are between quantum mechanics and the hamiltonian formulation of classical mechanics
@huy: looking at the Wilcox paper, the method of proof is to show that both sides are solutions to some appropriate inhomogeneous ODE with initial condition
which isn't a bad approach at all; i like that kind of method when doing BCH calculations
When I was younger and at university my professor told me how he got only top grades throughout his studies. I replied to that and said "Impressive" without really acknowledging that I could not have done the same.
It was very foolish of me to cut the discussion short like that. I should have listened to my professor more. Being overly positive in a such situation is actually wrong. It is a kind of a arrogance or narcissism to do that. I can't really express this thought clearly. But I think what I am trying to say is that if your peer tells you that he got only topgrades like he did, and …
he moreover asserts that the identity we're interested in can be then obtained by taylor expanding $H(\lambda+\epsilon)$ in powers of $\epsilon$ and identifying the first-order term
which is a neat idea, but not one i can verify at a glance
@robjohn Mathematica has a very hard time with this one Integrate[Log[Sin[x] + Cos[x]] Log[Sin[x]], {x, 0, Pi/2}]
Again, I'm so happy for Simona Halep, she suffered a lot when she lost some important games last months. Even some of our papers talked against her. Anyway, who works hard will finally win and break all the limits!
By the way, the previous question can lead us to another very nice integral ...
Well, yeah, I was disturbed by that paper I received that was suggested to be related to my proof of Au-Yeung series, that one changed my mood and that's why I said many things after that. The next time I'll have another achievement I won't make it known here (that's sure - to avoid any other such a discussion).
@UserX Because I work on ... $$\int_0^{\pi/2} \log\left(\sin\left(x+\frac{\pi}{4}\right)\right) \log(\cot(x)) \ dx$$ (this is why I'm quiet --- well, not that quiet)
These are specific cases of a couple of simple formulas I derived without using contour integration. Mathematica has difficulty approximating both integrals for specific values of $p>0$, especially the first one which only converges condition…
@RandomVariable No, but I can say they look awesome ...
@RandomVariable Have you ever considered to write a book of integrals, series and limits you discovered? I strongly believe it would be very appreciated and I'd like to have it.
If there is a thing that shock me to a certain extent is that I cannot manage to find such integrals in some books on market (with some exceptions --- well, the integrals in the books I read were not that complex)
@Max Yes; for write an arbitrary element of $I$ as $f(x) = \sum a_i(x)g_i(x)$. Then $f(x^n) = \sum a_i(x^n)g_i(x^n) \in \langle g_1(x^n), \dots, g_m(x^n)\rangle \subset I$.
@robjohn I'm not sure I get that point. Are they the same? hmmm, let me check again ...
@robjohn I saw that expansion in one answer of yours.
@RandomVariable Well, maybe it's time to think of it. People really need such books. It's a pity not to have your own book, since as far as I know you also have a lot of stuff.
@RandomVariable I said to @robjohn the same thing some time ago. He should definitely have a book too.
I believe this: a hand of people can change the face of the world by their actions (in a good sense). Sometimes we miss the proper books to see the beauty of things around us, the books that make us to fall in love with things.
@RandomVariable I recommend you "Limits, Series, and Fractional Part Integrals" by Ovidiu Furdui, from cover to cover, you'll have a lot of fun, there are very nice questions with very nice solutions.
@Chris'ssis If I ever get to a point where I'm more confident in my mathematical abilities, I might consider it. But right now I feel like there is way too much that I don't know. I still ask silly questions at times.
Before you answer this OP, please read all the terms and conditions below. Thank you...
Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for fun guys. (>‿◠)✌
Before we start the contest, here are the rules of my little contest that you ...
In 1638 Galileo published Two New Sciences, in which he described his inclined plane experiment.
He discovered that the acceleration of gravity was uniform, and could be modeled mathematically by the simple equation < Distance = c * t² >.
Question: Was this the first discovery of the fact ...
