Wikipedia:Reference desk/Archives/Mathematics/2010 May 23

= May 23 =

Help for an estimate
Hello, I would like to estimate several derivatives wrt the variable s of the integral function
 * $$I(s,a)=\int_{a}^1\sqrt{a^2 s+z^2}dz$$

I am looking for estimates of the form
 * $$| \frac {\partial^k I}{\partial s}(s,a)|\leq C$$

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.--pm a 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
 * $$a^2 s\int_{s^{-1/2}}^{s^{-1/2} a^{-1}}\sqrt{1+u^2}du$$
 * 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

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

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. --pm a 04:12, 24 May 2010 (UTC)