How can I find out wheter the function below is an analytical function or not?
$$x+\frac{x}{x^2+y^2}+i\left(y-\frac{y}{x^2+y^2}\right)$$
I found a solution with Cauchy-Riemann equations. However, I don't know that yet. How do I find out without Cauchy-Riemann?
Well,with $z=x+iy$,
$$x+iy=z \quad \text{ and } \quad \frac{x-iy}{x^2+y^2}=\frac{\overline{z}}{|z|^2}=\frac{1}{z},$$
So your function is $$f(z) = z+\frac{1}{z}.$$
Your function, $f(z)$, is holomorphic and thus analytic everywhere other than at the origin: $$z \in \mathbb{C} \setminus {0}.$$
One reason is because sums, differences, products, and quotients (other than when the denominator is zero) of holomorphic functions are holomorphic.