Find all points where the function $f(z) = 2x − 3iy$ is
differentiable.
I found the partial derivatives: $u_x=2$, $u_v=0$, $v_x=0$, and $v_y=-3$ and they do not satisfy the Cauchy-Riemann equations. Does this mean that $f(z)$ is not differentiable at any point?
Best Answer
The function is deffirentiable as a two real variables function everywhere since it is a polynomial. The fact that the Cauchy-Riemann equations are not verified anywhere means that the function is not holmorphic anywhere. Indeed $f$ is holomorphic in $z_0$ if $f$ seen as a two real variables function is differentiable in $z_0$ AND the Cauchy-Riemann equations are verified in $z_0$.