Harmonic Function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:44:51Z http://mathoverflow.net/feeds/question/112078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112078/harmonic-function Harmonic Function? hapchiu 2012-11-11T15:05:28Z 2012-11-11T15:05:28Z <p>Hi,</p> <p>Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + iy_{1}, ..., x_{n} + iy_{n})} \sqrt{\varphi (\xi)} d \xi)$ harmonic. Where integration should be understood: integrate along any path from $0$ to $(x_{1} + iy_{1}, ..., x_{n} + iy_{n})$ (by Cauchy the integral is path independet). With harmonic I mean: $\Delta u = \frac{\partial^{2}u}{\partial x_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial x_{n}^{2}} + \frac{\partial^{2}u}{\partial y_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial y_{1}^{n}} = 0$. I was reading about this and in 2 dimensions it is harmonic. Now I am actually interested if its also harmonic in more dimensions. Is it? </p> <p>hapchiu</p>