Existence of a special holomorphic function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:26:49Z http://mathoverflow.net/feeds/question/63784 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63784/existence-of-a-special-holomorphic-function Existence of a special holomorphic function J. Fabian Meier 2011-05-03T08:40:46Z 2011-05-03T09:11:00Z <p>How can you prove the existence of a nonzero function from the subset $U= \{z| 0 \leq Re z \leq 1\}$ of $\mathbb C$ to $\mathbb C$ which is holomorphic on the interior of $U$ and vanishes on the right boundary of $U$ ?</p> http://mathoverflow.net/questions/63784/existence-of-a-special-holomorphic-function/63785#63785 Answer by Denis Serre for Existence of a special holomorphic function Denis Serre 2011-05-03T09:11:00Z 2011-05-03T09:11:00Z <p>If you assume that your function is continuous on this right boundary (without that, your question should not make sense), then you can use a reflexion principle to extend its real part into a harmonic function $v$ in a neighbourhood of $z=1$. This harmonic function is the real part of a holomorphic function, thus your holomorphic function does extend across this boundary, as a holomorphic function $f$. This function has non-isolated zeroes, thus $f\equiv0$.</p> <p>Thus there is no solution to your problem.</p>