holomorhicity implies harmonic function in several variables I had read somewhere that it follows by cauchy riemann equations that any holomorphic or anti-holomorphic function  $f$ from  an open subset of  $C^n$ to $C$ is harmonic i.e $\sum_{i=1}^n \frac{\partial^2f}{\partial z_i^2}=0$.but i am not able to prove it ?Is it true?if yes,then how?
 A: You have transported the Laplace operator to $\mathbb{C}^n$ in the wrong way. Harmonicity is a real concept, the Laplace operator contains the partial derivatives with respect to the real coordinates. Writing $z_k = x_k + iy_k$, we then get
$$\Delta f = \sum_{k=1}^n \frac{\partial^2 f}{\partial x_k^2} + \sum_{k=1}^n \frac{\partial^2 f}{\partial y_k^2} = \sum_{k=1}^n \left(\frac{\partial^2f}{\partial x_k^2} + \frac{\partial^2 f}{\partial y_k^2}\right).$$
Now a small computation shows that in the Wirtinger derivatives
$$\frac{\partial}{\partial z_k} = \frac12\left(\frac{\partial}{\partial x_k} - i\frac{\partial}{\partial y_k}\right);\qquad \frac{\partial}{\partial\overline{z_k}} = \frac12\left(\frac{\partial}{\partial x_k} + i\frac{\partial}{\partial y_k}\right)$$
we have
$$\frac{\partial^2}{\partial x_k^2} + \frac{\partial^2}{\partial y_k^2} = 4\frac{\partial^2}{\partial z_k\partial \overline{z_k}},$$
and hence can write in complex form
$$\Delta f = 4\sum_{k=1}^n \frac{\partial^2 f}{\partial z_k\partial\overline{z_k}}.$$
In that form, it is evident that every holomorphic or antiholomorphic function $f$ is harmonic, since every term in the sum vanishes, as $\frac{\partial f}{\partial \overline{z_k}}= 0$ for all $k$ or $\frac{\partial f}{\partial z_k} = 0$ for all $k$ is the condition for holomorphy resp. antiholomorphy.
Thus a holomorphic or antiholomorphic function is not only harmonic, it is also harmonic in every variable separately, and more, it is pluriharmonic, its restriction to every complex line is harmonic.
