Which of the following function f(z), of the complex variable z, is NOT analytic at all the points of the complex plane?

The correct option is f (z) = log z.

To determine which function is not analytic at all points of the complex plane (i.e., not an entire function), let us analyze each of the given options:

1. $f\left(z\right)=z^2$:
This is a polynomial function. All polynomial functions are differentiable everywhere in the complex plane, meaning they are analytic (entire) everywhere.

2. $f\left(z\right)=sinz$:
The complex sine function is defined as:
$sinz=\frac{e^{iz}-e^{-iz}}{2i}$
Since the exponential function $e^z$ is entire, any linear combination of exponentials is also entire. Thus, $sinz$ is analytic everywhere.

3. $f\left(z\right)=e^z$:
The complex exponential function is differentiable at every point in the complex plane, making it an entire function.

4. $f\left(z\right)=logz$:
The complex logarithm function is a multi-valued function defined as:
$logz=ln\left|z\right|+iarg\left(z\right)$
This function is not defined at $z=0$ since $ln\left|0\right|$ is undefined. Furthermore, to make it single-valued and analytic, we must restrict its domain by introducing a branch cut (commonly along the negative real axis, from $-\infty$ to $0$). The function is not continuous, and therefore not differentiable or analytic, along the branch cut and at the branch point $z=0$.
Consequently, $f\left(z\right)=logz$ is not analytic at all points of the complex plane.