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.
Best Answer
This function fails to satisfy the Cauchy-Riemann equations and is is therefore not complex-differentiable.