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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?


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Even in dimension $n=1$ your integral is not defined if $\phi(z)=0$ at some point. –  Alexandre Eremenko Nov 11 '12 at 20:22
why so? can you explain? –  hapchiu Nov 12 '12 at 4:47

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