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I understand your proof and the closure of $$\bar{G}$$ follows quite easily just by multiplying the matrices, however I'm not quite convinced on the solution of this exercise. I'm quite grateful on your help, but I think I need to study some prerequisites of this book and in mathematics in...

I will think about your post and reply here later, thank you very much.

So you use that the closure of a set is itself plus its boundary? Let me explain it myself to see if I'm understanding this: Because $$G \subseteq \bar{G}$$ we only need to compute the boundary part. Now we want to see if the number $$\frac{a(t+2\pi n)}{s+2\pi m } << 1$$ but I can't put that...

Let $$GL(2;\mathbb{C})$$ be the complex 2x2 invertible matrices group. Let $$a$$ be an irrational number and $$G$$ be the following subgroup $$G=\Big\{ \begin{pmatrix}e^{it} & 0 \\ 0 & e^{iat} \end{pmatrix} \Big| t \in \mathbb{R} \Big\}$$ I have to show that the closure of the set $$G$$...

You may find this link interesting Surya's Blog!: Finding Rank of a Word: With Repetition

I've only read half topic but it has an insane amount of advice, references, and enjoyable stuff. Thank you all, seriously.

Homework Statement Show that if z_1,z_2 \in \mathbb{C} then |z_1+z_2| \leq |z_1| + |z_2| Homework Equations Above. The Attempt at a Solution I tried by explicit calculation, with obvious notation for a,b and c: my frist claim is not that the triangle inequality holds, just that...

To be sure if I understood this, my answer to the first integral is $i\pi^3/4$. If $c=\{2+e^{i\theta} : \theta \in [0,2\pi] \}$ as the poles are all in the imaginary axis, and $c$ is the circle of radius $1$ and center $2$ it never touches the imaginary axis, therefore no poles inside $c$, so...

Hi! I'm wondering how do we study mathematics. If the book has exercises, after reading (maybe several times) the theory, you can go and do the exercises. If the book is only theory and no exercises, how do you check your understanding on the subject?

Thank you, now I get it! I didn't know the $\varepsilon(z)$ thing, my math training is the one I am being given in my physics undergraduate courses so it's quite mechanical and we don't care too much about math rigor. Such a bad mistake in my opinion, but I'm my spare time, which is not as much...

Hi, thanks for the hint! Ok, now I see it, after some google research, that your result for the pole is the same as L'Hôpital's rule $$\lim_{z\to p}(z-p)\frac{g(z)}{h(z)}=\lim_{z \to p}\frac{g(z)}{\frac{h(z)}{z-p}}=\lim_{z \to p}\frac{g(z)}{\frac{h(z)-h(p)}{z-p}}=\lim_{z \to...

Hi, thanks for your reply. The residue is: $$\lim_{z\to i\pi/2}\left(z-\frac{i\pi}{2}\right)\frac{z^2}{e^{2z}+1}$$ Now, the "problem" is that I don't know to "factorize" $e^{2z}+1=0$, to cancel with $\left(z-\frac{i\pi}{2}\right)$ I was thinking in circular and hyperbolical functions, but...

Hi all! I have to perform this complex integration over three curves, the first one is \( C=\{ z \in \mathbb{C} : |z|=2 \} \) and the function to integrate is $$ f(z)=\frac{z^2}{e^{2z}+1}$$ If I do the usual change of variables \(z=2e^{i\theta} \) and integrate from \( \theta = 0 \rightarrow...

The right to think about, I guess, it's that when you are here or here, you measure some time. Every observer has some rules and some clocks to measure time. Now suppose the twin paradox. The thing that actually happens is that their clocks will not measure the same amount of time, but time...

I know, in fact i should do it for my speclal relativity exam, but there is some little work and theory before Lorentz Transformations.