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May 23

Help for an estimate

Hello, I would like to estimate several derivatives wrt the variable s of the integral function

$I(s,a)=\underset{a}{\overset{1}{\int }}\sqrt{a^{2}s+z^2}dz$

I am looking for estimates of the form

$|\frac{{\partial }^{k}I}{\partial s}(s,a)|\le C$

where C does not depend on a. In principle one could estimate every single derivative but it would take a very long time and so I am looking for a general scheme.

Any idea?--Pokipsy76 (talk) 16:34, 23 May 2010 (UTC)

As a firs step, I'd change variable and write $z:=a|s{|}^{1/2}u.$ Not clear what's the range of a and s, btw.--pma 17:06, 23 May 2010 (UTC)

Thank you for your help. Your suggestion is very smart (I don't think I could ever think it) indeed the integral becomes

$a^{2}s\underset{s^{-1/2}}{\overset{s^{-1/2}{a}^{-1}}{\int }}\sqrt{1+u^2}du$

so when I derive it wrt s I obtain a simple algebraic expression which can be estimated easily. It was not obvious to me that having the variable only on the integration domain would allow to eliminate the integral after a derivative.--Pokipsy76 (talk) 17:38, 23 May 2010 (UTC)

Isometries of the Poincare Disc

Hi everyone,

Could anyone direct me to a proof of the fact that orientation-preserving isometries of the hyperbolic disc are of the form

$\psi \to w\frac{\psi -\lambda }{\overline{\lambda }\psi -1}$

where $|\lambda |<1$ and $|w|=1$?

I have the result but I can't seem to find a proof anywhere online. Of course, if you could provide a short proof instead that would be greatly useful too, but I'm very much capable of understanding a proof on my own if it'll save you time linking be to one rather than writing it out yourself :-)

Thanks very much,

82.133.94.184 (talk) 23:05, 23 May 2010 (UTC)

The characterization of the holomorphic automorphisms of the disk is in almost every book on complex variable, e.g Rudin's Real and complex analysis. Here there is something in Möbius transformation. --pma 04:12, 24 May 2010 (UTC)