When $g(x,y)= x(x^2+y^2)^{p/2} $ is real analytic at origin? A complex valued function $f,$ defined on  an open set  $E$ in the plane $\mathbb R^{2}$, is said to be real analytic on $E$, if to every point $(s_{0}, t_{0}) \in E,$ there corresponds an expansion of the form
$$F(s, t)= \sum_{m,n=0}^{\infty} a_{mn} \, (s-s_{0})^{m} \, (t-t_{0})^{n}, \hskip.1in a_{mn} \in \mathbb C$$
which converges absolutely for all $(s,t)$ in some neighbourhood of $(s_{0}, t_{0}).$

Let $g:\mathbb R^2\to \mathbb R$ such that $g(x,y)=x(x^2+y^2)^{p/2}$ for $p\in (0, \infty).$
My Question: (1) Can we prove that $g$ is not  real analytic at $(0,0)$ if $p\in (0, \infty)\setminus 2 \mathbb N$? (2) What can we say if $p\in 2 \mathbb N$.
Let $f:\mathbb R^2\to \mathbb C$ is defined by
$$f(x,y)= x(x^2+y^2)^{p/2} + iy(x^2+y^2)^{p/2}$$  where $p \in (0, \infty)$

My Questions: (1) Can we prove that $f$ is not  real analytic at $(0,0)$ if $p\in (0, \infty)\setminus 2 \mathbb N$? (2) What can we say if $p\in 2 \mathbb N$.

Rough Thoughts:  Real analytic functions are infinitely many times differentialble. Maybe to prove (1) I need to show that  $f$ fails to be differentiable (after some derivaties! ) at $(0,0)$?

Your function has a branch point at the origin for all cases that $\frac{p}{2}$ is a noninteger, and is therefore not analytic at the origin
