Kannaguchi O.
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 Jul 2 awarded Curious Nov 30 comment Is there a conformal self-map on the upper half-plane that swaps two points? Oh, right, it may be better to think about this on the unit disk since it and the upper half-plane are isomorphic. I'll see what I can cook up, thanks. Nov 30 asked Is there a conformal self-map on the upper half-plane that swaps two points? Nov 18 awarded Commentator Nov 16 comment Möbius map from circles to lines I tried the second question using the cross ratio, but it doesn't tell me much, just that the ratio is real (which we already knew). I'm thinking now that I want to prove there is a line that that the four points are symmetric in pairs with respect to. Then if I map that line to the imaginary axis, I'll get symmetric points $\pm a, \pm b$ which I can scale down as appropriate. Nov 16 accepted Möbius map from circles to lines Nov 16 comment Möbius map from circles to lines That's what I did. Before, I tried mapping the imaginary axis to the real axis and $-2 \mapsto \infty$, since this would be the desired configuration, by $4i \mapsto 1, 0 \mapsto 0$. This didn't work, though. Turns out I had to do $4i \mapsto 0, 0 \mapsto 1$, which, I guess, had the same orientation as the original circles. Nov 16 revised Möbius map from circles to lines added 158 characters in body Nov 16 asked Möbius map from circles to lines Nov 11 revised Alternate form of the Residue theorem? added 20 characters in body Nov 10 asked Alternate form of the Residue theorem? Nov 9 accepted Rewrite a series as a power series? Nov 9 asked Rewrite a series as a power series? Oct 20 accepted Why is the intersection of a clopen set and another set clopen in this proof? Oct 20 comment Why is the intersection of a clopen set and another set clopen in this proof? Oooooh, sorry, I misread. Any subset is clopen in its subspace topology, but certainly not in the subspace topology of $W_{\alpha_0}$. I managed to confuse myself with subsets and subspaces. Oct 20 comment Why is the intersection of a clopen set and another set clopen in this proof? I thought that might be the case, but isn't any subset of $W_{\alpha_0}$ clopen in the subspace topology? Oct 20 asked Why is the intersection of a clopen set and another set clopen in this proof? Oct 1 awarded Scholar Oct 1 accepted Find entire functions that satisfy certain conditions Oct 1 comment Find entire functions that satisfy certain conditions In the first that inequality holds for any $z$ by uniform continuity. But since the LHS holds for any smaller choice of delta, it holds in the limit as $\delta \to 0$. So I concluded that the derivative was bounded, though now I see that your comment above concerned this very conclusion and I haven't used LVK's fact about that difference being entire. With the integral, by induction and the Gauss MVT I conclude that $f^{(n-1)}(0) = 0$ for all n. So by uniqueness, it must be identically zero on the unit disk, and therefore zero on the entire plane.