User morris kalka - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:34:55Z http://mathoverflow.net/feeds/user/7436 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77120/smoothness-of-f-sqrt-x/77161#77161 Answer by Morris Kalka for Smoothness of $f(\sqrt x)$ Morris Kalka 2011-10-04T18:27:33Z 2011-10-04T18:27:33Z <p>Apparently this fact was first proved by Hassler Whitney and can be found in the Duke Journal 1943, volume 10 no 1 p.159-160 under the title Differentiable Even Functions.</p> http://mathoverflow.net/questions/43484/an-analytic-proof-of-the-de-franchis-theorem/62568#62568 Answer by Morris Kalka for An analytic proof of the De Franchis theorem Morris Kalka 2011-04-21T20:09:31Z 2011-04-21T20:09:31Z <p>We break the proof up into two steps: 1. The set of holomorphic maps is compact in the compact open topology. This is an easy consequence of the hyperbolicity of the Riemann surfaces</p> <ol> <li>The space of holomorphic maps is discrete. Suppose that there is a one parameter family (f_t) of maps between the Riemann surfaces. The derviative (\frac{\partial f}{\partial t}\0 is then a holomorphic section of the pull-back of the tangent bundle of the range over the domain. By hyperbolicity there are no non-zero sections. </li> </ol> <p>You can find arguments of this type in</p> <p>M. Kalka, B. Shiffman and B. Wong, Finiteness and Rigidity Theorems for Holomorphic Mappings, Michigan Math. J. 28 (1981), 289-295.</p>