Hi there. I have to study the analyticity for the complex function:

The exercise suggest me to use polar coordinates. So, I do this kind of exercise using a theorem that says that if the function acomplishes the Cauchy-Riemann conditions, and the partial derivatives are continuous in the vecinity of a point, then its analytical in that region.

At first glance I would say its analytical over the entire complex plane except at the point zero, because its a division between polynomials, and the denominator is zero at zero. But I tried to demonstrate it as the exercise suggests me using polar coordinates.

So this is what I did:

Then

And

The Cauchy-Riemann conditions in polar coordinates are

And for my function I got:

So to acomplish Cauchy Riemann I should get:

And

Then it isn't analytical over the entire complex plane, which contradicts the assumption that I've made at first, so what did I do wrong?

PD: I didn't know where to post complex analysis question, so I'm not sure this is the propper section, if its not, please move it. Thanks.