@PeterTamaroff there where some time a lot of exercises about the Cantor set and the Cantor-Lebesgue function. Those are the kind of things that in books are assumed true (because they are) and are then exercises to the reader. For example I've searched for a detailed construction of the Cantor set. I mean not just pictures. I found nothing at that time. So tried to write it myself, when finished, it seemed nice to me. Slick. So that's why I wanted to post it in somewhere
In a blog, there is no restriction to the pages you write for example. Because there is no pages, no paper
you can put all the equations, pictures, whatever you want
there where some other things. For example the fact that the complex numbers are a subset of the group of linear operators in $\Bbb R^2$. That multiplication by of a complex number by another one is again another such linear operator. That the complex derivative induce a differential which is again another linear operator. All this gives for free the Cauchy-Riemann equations for example
mathematicians all know that, but they never state it clearly
the generic explicit matrix-form solution to the system involves a definite integral, so there's no a priori ODE-related-reason to expect the expression to simplify, if that's what you're wondering about
it seems to me the non-elementary trigonometric special functions involved so far in the thread are inescapable one way or another in explicit solutions, so settling looks like the best option
hmm, I want to see how L/K Galois implies primes (of L) lying over a given one of K have equal ramification indices
I found this answer, outlining the exercise, to be interesting. However, I have trouble solving the differential equation.
The question starts by attempting to solve the following integral without complex analysis:
$$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
...So we let
$$ F(y) = \in...
This was a question on an old prelim exam in complex analysis: compute
$$\int_0^\infty \frac{\cos(tx)}{x^2 - 2x + 2}\,\mathrm{d}x$$
for $t$ real. I've tried…
Residue calculus—it's easy to integrate the similar $\int_0^\infty \frac{\cos(tx)}{x^2+2} \mathrm{d}x$ largely because the integrand is...
I always bring a buldozer when planting flowers, and dynamite. And a small angry dwarf with a harpoon. Just in case, you may never know when you will need him.
@N3buchadnezzar it's not hard, but there is some work to do that involves at some points beta function and digamma function (my guess by only looking at it).
@robjohn =) I was unsure about that part. Do you have a counterexample in mind? Anyway using $R(x) = 1/x$ does not converge. So it seems it is not enough that $R$ satisfy the differential equation.
@N3buchadnezzar I'm thinking to apply Cauchy d'Alembert to $\lim_{n\to\infty} \left(\int_0^n 1+\arctan^2(x) \ dx\right)^{1/n}, \space n \in \mathbb{N}$, but not sure what to do next.
hmmm, I think I should create some more limits of this type.
@robjohn The previous one was: $" \text{If} \lim_{x\to\infty} (x^2 f'' (x) + 4 x f' (x) + 2 f (x)) = 1, \text{then compute} \lim_{x\to\infty} x f'(x)"$
@N3buchadnezzar Did you see my answer? The integral from $0$ to $\infty$ is messy with special functions. The integral from $-\infty$ to $\infty$ is much nicer.
@Chris'swisesister does your browser show you where the arrows point in the comments?
I always wonder who first came up with various beautiful proofs and identites.
I mean looking at a integral, introducing a seemingly arbitary constant. And showing that this function satisfy some odd looking differential equation...
@Chris'swisesister at the left of this reply should be a little arrow. That arrow means that this reply is in response to another. That is the one that is highlighted when you hover over this one. If it is off the page, just click on the arrow.
@robjohn for $\lim f(x)$ I think one way that would work great is to write $\lim x^2 f(x)/x^2$, and then apply l'Hopital and use the initial condition.
Integrating by parts, $$ \begin{align} \int_a^bxf'(x)\,\mathrm{d}x&=bf(b)-af(a)-\int_a^bf(x)\,\mathrm{d}x\\ \int_a^bx^2f''(x)\,\mathrm{d}x&=b^2f'(b)-a^2f'(a)-2\int_a^b xf'(x)\,\mathrm{d}x\\ &=b^2f'(b)-a^2f'(a)-2bf(b)+2af(a)+2\int_a^bf(x)\,\mathrm{d}x\\ \int_a^b\left(x^2f''(x)+4xf'(x)+2f(x)\right)\,\mathrm{d}x &=b^2f'(b)+2bf(b)-a^2f'(a)-2af(a) \end{align} $$ For $a,b$ large enough so that $\left|\,x^2f''(x)+4xf'(x)+2f(x)-1\,\right|\le\epsilon$, this means that $$ \left|\,\left(b^2f'(b)+2bf(b)-b\right)-\left(a^2f'(a)+2af(a)-a\right)\,\right|\le\epsilon|b-a|
Integrating by parts again, $$ \begin{align} \int_a^bx^2f'(x)\,\mathrm{d}x&=b^2f(b)-a^2f(a)-2\int_a^bxf(x)\,\mathrm{d}x\\ \int_a^b\left(x^2f'(x)+2xf(x)\right)\,\mathrm{d}x&=b^2f(b)-a^2f(a) \end{align} $$ For $a,b$ large enough so that $\left|\,xf'(x)+2f(x)-1\,\right|\le\epsilon$, this means that $$ \left|\,\left(b^2f(b)-\frac12b^2\right)-\left(a^2f(a)-\frac12a^2\right)\,\right|\le\epsilon\,\left|\,\frac12b^2-\frac12a^2\,\right| $$ Therefore, we get that $$ f(x)\to\frac12\tag{2} $$ Combining $(1)$ and $(2)$ yields
@Chris'swisesister I think the fact that $f(x)\to\frac12$ seems more interesting than $xf'(x)\to0$, but you do need to show $f(x)\to\frac12$ first, I guess.
If l'Hopital is allowed, then $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{x^2 f(x)}{x^2}=\frac{1}{2} $$ where the last equality comes from the initial condition.
That's because $(x^2 f'' (x) + 4 x f' (x) + 2 f (x)) = (x^2 f(x))''$
Maybe the version of the problem I received is not complete.
@robjohn We can write that $\lim_{x\to\infty}(x f'(x)+f(x)-f(x))=\lim_{x\to\infty} ((x f(x))'-f(x))$. If l'Hopital is allowed $1/2=\lim_{x\to\infty} f(x) =\displaystyle\lim_{x\to\infty} \frac{x f(x)}{x}= \displaystyle\lim_{x\to\infty} (x f(x))'$. Hence $\lim_{x\to\infty} x f'(x)=1/2-1/2=0$
Quick opinion poll: is a 'where can I go to learn more about X now' question appropriate for math.SE? In my case, it's a subject where there's one wonderful semi-casual reference and the rest of the material seems deathly dry; it may be a bit of a dead subject, but I'm curious if there's another good work that's a step up from what I've read.
@Chris'swisesister one thing slapped me in the face while I was getting ready to leave: L'H says that if $f(x)$ and $g(x)$ go to $\infty$ and if the limit of $f'/g'$ exists, then the limit of $f/g$ is the same. It doesn't work backwards.
@skullpatrol No, I was just getting ready (showering, etc)
If you are given $3$ points and are told to draw a circle through those points, the circle is uniquely defined. Given $4$ points and an ellipse (I think) the ellipse is uniquely defined. What analogous shape (i.e. one that ellipses and circles are special cases of) requires $5$ points to uniquely define it?
But the whole thing is idiotic, because people insist on statistical certainty using data taken over a long period of time, then expect that to have meaningful near-term results.
As if somehow, the events that happened when a player was 27 but had a sprained shoulder will have an equal effect on tomorrow's game as that same player's recent stats as a 33 year old after starting a new workout regimen.
@PeterTamaroff $Fr(A)$ is the set of frontier points of $A$ where a frontier point is any point whose neighborhood intersect both $A$ and $\mathcal{R^n}-A$. The def. you are using is another theorem that I have to prove.
A set $A$ is closed if there exists a neighborhood $N$ of every $x\notin A$ such that $N\cap A=\emptyset$
@saadtaame Note you define a closed set as $x\notin F\implies \exists N_x:N_x\cap F=\varnothing$. Thus $\forall N_x:N_x\cap F\neq \varnothing\implies x\in F$ (this is the contrapositive of the first)
@Chris'swisesister and that is not the converse of L'Hospital; that is the usual version. If you know the limit of the ratio of the derivatives, you know the ratio of the functions.
Is there a quick way to functional analysis (Hilbert spaces)? Or do you generally have to have a really good understanding of analysis (say Rudins principles or more)? In particular I was thinking about doing something with studying infinite Cayley graphs using (analogs of) adjacency matrices with the hopes of learning/applying to group growth functions.
This was a question on an old prelim exam in complex analysis: compute
$$\int_0^\infty \frac{\cos(tx)}{x^2 - 2x + 2}\,\mathrm{d}x$$
for $t$ real. I've tried…
Residue calculus—it's easy to integrate the similar $\int_0^\infty \frac{\cos(tx)}{x^2+2} \mathrm{d}x$ largely because the integrand is...
What are the tags reference-request, soft-question, homework, big-list for? When are they used? When should they not be used?
