I have the following function and I am trying to find if it is analytic and differentiable. I use cauchy-riemann to prove it.
$$ f(x) = x^2 -x+y+i(y^2-5y-x)$$
$$u(x,y) = x^2-x+y$$ $$v(x,y) = y^2-5y-x$$
$$u_x = 2x-1$$ $$u_y = 1$$ $$v_x= -1$$ $$v_y= 2y-5$$
As a result $$u_y = -v_x \Rightarrow 1 = -(-1) \Rightarrow 1 = 1$$ and $$u_x \neq v_y\Rightarrow y = x+2$$
I was wondering if we can say that there some regions that the function is differentiable or analytic.