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.