Answer :

Solution :

The complex function f(z) = u (x, y) + iv (x, y) is to be analytic if it satisfy the following two conditions of Cauchy-Reiman theorem.

- $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}and\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

- $\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{x}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{y}.$

f(z) = log z

$\frac{\partial }{\partial z}f(z)=\frac{1}{z}$

Here, the function f(z) is analytic except z = 0. Since the function is not defined for these two values.

But in question it is asked at all points hence f(z) = log z is not analytic at all points.