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Emolga

 Mathematics

Associated with Math.SE; for both general discussion & math qu...
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Emolga
Feb 20, 2021 19:26
lol didn't know that either
Emolga
Feb 20, 2021 19:26
Yes, I now realize that
Emolga
Feb 20, 2021 19:24
@Astyx @MikeMiller I understand now, thank you very much!!
Emolga
Feb 20, 2021 19:07
I agree that $B(x-\lambda I)=x^n-\lambda ^n I$
Emolga
Feb 20, 2021 19:05
The book claims that if $\lambda \in \sigma (x)$ then $\lambda ^n \in \sigma (x^n)$
Emolga
Feb 20, 2021 19:05
I am not happy with this because in reality I need it in a general Banach algebra
Emolga
Feb 20, 2021 19:01
So it's not clear to me why (non-invertible) times (something) cannot be invertible
Emolga
Feb 20, 2021 19:00
Mike, I don't assume finite dimensional
Emolga
Feb 20, 2021 19:00
I know that $T-\lambda^n I$ is of the form $T-\lambda I$ times something
Emolga
Feb 20, 2021 18:59
Why if $\lambda$ is in the spectrum of some continuous linear transformation $T$ then $\lambda ^n$ is in the spectrum of $T^n$?
Emolga
Feb 10, 2021 19:42
Thank you
Emolga
Feb 10, 2021 19:40
really?
Emolga
Feb 10, 2021 19:40
because I can't prove this
Emolga
Feb 10, 2021 19:39
@Thorgott Yes, although I need the very concrete case of a holomorphic covering map between two domain. Is it true that its derivative cannot vanish?
Emolga
Feb 10, 2021 19:37
Can a covering map have a critical point?
Emolga
Aug 1, 2020 21:34
How do we go from areas to lengths?
Emolga
Aug 1, 2020 21:33
("area preserving and conformal forces isometry"?)
Emolga
Aug 1, 2020 21:03
And why are there no more hyperbolic isometries except those?
Emolga
Aug 1, 2020 20:53
Maybe it is, if we start from saying hyperbolic space is where distance is a conformal invariant. So hyperbolic space is where maps preserve angles iff they preserve distance.
Emolga
Aug 1, 2020 20:52
Is that "expected"?
Emolga
Aug 1, 2020 20:41
a function in the disk is conformal iff it is a hyperbolic isometry
Emolga
Aug 1, 2020 20:41
I mean, there's an iff
Emolga
Aug 1, 2020 20:41
But what I wrote is rather exact...
Emolga
Aug 1, 2020 20:40
I figured it's involved enough to warrant a post, so I work on phrasing it now
Emolga
Aug 1, 2020 20:26
Why it "makes sense"?
Emolga
Aug 1, 2020 20:26
Is there a philosophy why an angle-preserving Euclidean map corresponds to a distance-preserving hyperbolic map?
Emolga
Jul 29, 2020 21:54
(I actually saw exactly this once)
Emolga
Jul 29, 2020 21:54
Pretty sure it works
Emolga
Jul 29, 2020 21:44
I posted your answer in the post. Thank you!
Emolga
Jul 29, 2020 21:41
I think you solved it, Ted
Emolga
Jul 29, 2020 21:41
Oh, cool
Emolga
Jul 29, 2020 21:40
isolated is not a problem with accumulating to the boundary. Continuity probably is.
Emolga
Jul 29, 2020 21:40
So what do you mean?
Emolga
Jul 29, 2020 21:39
But Blaschke just moves points around in the disk
Emolga
Jul 29, 2020 21:39
I guess you really can't because of continuity
Emolga
Jul 29, 2020 21:36
so it is a pole
Emolga
Jul 29, 2020 21:36
Really? but it is the inverse of a holomophic function at 0
Emolga
Jul 29, 2020 21:35
Posted
Emolga
Jul 29, 2020 21:35
0
Q: Is a meromorphic function determined by its boundary values?

EmolgaLet $f: \mathbb D \to \widehat {\mathbb{C}}$ be a meromorphic function inside the unit disk. Assume that $f$ is zero on the boundary and continuous in the closed disk (as a function into $\widehat {\mathbb{C}}$). Is $f$ necessarily identically zero? If $f$ is not surjective then we can take $a\...

complex-analysis
Emolga
Jul 29, 2020 21:33
Ok, since it turns out to be not quick I'll post it in a question
Emolga
Jul 29, 2020 21:31
Oh, I take it back
Emolga
Jul 29, 2020 21:30
it has a singularity there
Emolga
Jul 29, 2020 21:30
It is not continuous at 1
Emolga
Jul 29, 2020 21:28
Can I?
Emolga
Jul 29, 2020 21:22
I don't know if there exists a surjective example
Emolga
Jul 29, 2020 21:09
What if it's surjective?
Emolga
Jul 29, 2020 21:06
If the function is not surjective then we are done: choose a value $a$ not in the image of the function, then consider $1/(f(z)-a)$.
Emolga
Jul 29, 2020 21:01
Not that I see
Emolga
Jul 29, 2020 20:59
Let $f$ be a meromorphic function in the unit disk that is zero on the boundary and continuous in the closed disk (as a function to the Riemann sphere). Does it follow that $f$ is identically zero?
Emolga
Jul 29, 2020 20:58
Yes, but this suppposes that we are holomorphic in a neighborhood of the disk