I think I know how to prove essential part, by the definition provided by textbook. A singularity point is essential when it is neither equal to infinity or bounded. Hence, it is equivalent to say that limit does not exist at that singularity point because left side limit and right side limit are different. But for analytic part, I try to solve it by Riemann Cauchy equation, but the function itself is very complicated by separating u and z to a+bi and x+yi.

Hence, I wonder if we can have a better way to prove analytic part or I just need to keep calculating.