
Analytical function
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 CauchyRiemann 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 CauchyRiemann 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.

Re: Analytical function
Your assumption is wrong. This is NOT "a division between polynomials" because in functions of complex variables a "polynomial" is specifically a polynomial in the variable z, not in the real and imaginary parts of z.

Re: Analytical function
If then is...
(1)
... so that f(z) isn't analytic in any point of the complex plane...
Kind regards

Re: Analytical function