Well tell me something if i have been given a matrix of a linear transformation and of what it does to the standard order basis how can i work back to get the linear transformation?
@Hippalectryon here is an example $$\int_{-\infty}^{\infty} \frac{e^{x+1}+1}{e^x-1} \cdot \frac{1}{\pi ^2+(x+1)^2} \ dx=\frac{1+e}{1+\pi ^2}$$ You don't want to rely on Mathematica here. :-)
@Hippalectryon The amazing thing about the question above is that I finalized it by real analysis only (this is crazy hard - but pretty easy by complex analysis) :D
@Hippalectryon It's hard to know what to do, how to make the connections, but the proof is very clean, short, fast, elegant, easy to understand (that was possible in my case due to some personal research).
Would like some help setting up a problem calc problem. When I do this problem, I get half of the correct answer. I'm looking for the volume of a solid with semicircular base of radius 9 and cross section "perpendicular to the base and parallel to the diameter" are squares. So the sides of each square are $y=\sqrt{81-x^2}$ and the area of each square is $A(x)=81-x^2$. When I integrate that from -9 to 9 I get $V=\int_{-9}^9 (81-x^2) dx = 972$. The correct answer is 1944. What am I doing wrong?
I never really became proficient with solving such problems by slicing, but $4\int _0 ^9 \int _0 ^{\sqrt{81-x^2}} \int _0 ^{\sqrt{81-x^2}} \mathrm{d}z \, \mathrm{d}y \, \mathrm{d} x$ gives the right answer
the product of the integrals gives $\frac{1}{4}$ the volume of the solid: you integrate from the xy-plane, z=0, to the height of the square cross-section, $z=\sqrt{81-x^2}$
@Chris'ssis my interest lie in the interplay between physics and math that is I want to do physics but on really rigorous way as mathematicians do math
@KarimMansour Ah, I see. There are some famous problems that arose in the physics, some of them related to integrals, series and limits - one example is the Ahmed integral.
@KarimMansour Many of them are very hard. I remember a series I posted some time ago here, it was proposed by Omran Kouba, and no one sent me a solution to it so far. I shared it with some students and professors.
@Chris'ssis that is interesting do you have like a blog or something I would love also getting to read your work too as such stuff will be valuable to me.
@KarimMansour Some of my latest research will allow an elementary solution to that problem I asked on main. I might add that to my book (not sure yet - I already exceeded the number of problems to add).
not so hard to prove straight from the definition when you get there; for now, take my word for it
so pick an open set $U$; its intersection with $(-n,n)$ $U_n$; then $f(U_n)$ is open for all $n$, and thus $f(U) = f(\cup U_n) = \cup f(U_n)$ must be open
@Hippalectryon If I did it, just a self-educated person, then someone with uni background and a lot of other courses combined with very hard work should do far better than me. :-)
@Hippalectryon Yeah, but after a while you can specialize in analysis, for example. Then you take a lot of helpful courses, meet a lot of experienced professors. Instead, when I have a problem I need to talk to myself most of the time.
so the idea is: $f$, restricted to $[-n,n]$, is a homeomorphism onto its image. so $f(U \cap (-n,n))$ is open in $f([-n,n])$ in the subspace topology. the only worry is that maybe this being open in the subspace topology is not actually ope in $\Bbb R$ itself. i think that's not possible but it seems like a little work to check
@Hippalectryon The art of making all needed mathematical connections to create a simple, marvellous picture with the solution is just wow. I cannot describe the amazing feeling you have. :-)
So the way I think to do it is pull the 2 out of $e^2$ and then set the exponents together. $-x=2x+2$ and got $x=-\frac{2}{3}$ which is the answer in the book
@Hippalectryon I'm also stuck sometimes, but after a while, when working on a different problem, an idea comes to mind and then know how to solve that problem. I never give up with the tough problems, I return to them after a while and solve them.
@Maximilian What is the difficulty there. Get in both sides $e^{something}=e^{something*}$ and then consider something=something* and from there you get the solution (explaining things in a non-mathematical way though). :-)
If $S$ is a subset of a topological space that is the union of an open dense set and complement of some other open dense set, then surely $S$ is not necessarily open right? Surely, even in places like $\mathbb R$?
consider $(\Bbb R - A) \cup B$ where $A$ is union of very small open intervals around the rationals in $\Bbb R$ and $B$ is, say, irrationals in $[0, 1]$ intersection $\Bbb R - A$.
well, what i said works for $\Bbb R^2$, say. leave $A$ as it is on the x-axis, and intersect the complement with a $B$ homeomorphic to $A$ away from the x-axis.
@Samuel: In all four books I've written, the thing I'm most proud of (and the thing Jasper doesn't care about) is the exercises. I made up a number of unusual ones.
@r9m The Knuth's problem is wrong. I was preparing to write the paper and thought to add that point to remark. Anyway, the problem is corrected by me as I think it should look like and then sent.
So, no concern about that.
It's good that I have the habit of checking the details numerically ... I hate the problems wrongly posed ...
I need to also explain the mistakes in the paper, since the only problem lies in the left side of the quality (generating function of the Catalan's number? Sorry? No)
Prove that
$$\sum_{k=1}^{\infty} \frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$$
