I have a homework question, and I'm having a hard time interpreting it.
Question: Where is the function $f(x+iy)=x^4y^5+ixy^3$ complex differentiable? Determine the derivative in such points.
My first plan was to find a region for which the following theorem applied:
Suppose $f=u+iv$ is a complex-valued function defined on an open set $\Omega$. If $u$ and $v$ are continuously differentiable and satisfy the Cauch-Riemann equations on $\Omega$, then $f$ is holomorphic on $\Omega$ and f'(z)=\frac{\partial f}{\partial z}.
Of course, $u$ and $v$ are going to be continously differentiable; the only question is, on what region are the Cauchy-Riemann equations satisfied?
So, I found that the Cauchy-Riemann equations in this case are the following:
$4x^3y^5=3xy^2$ and $5x^4y^4=-y^3$
and the only point at which these equations are satisfied is $(0,0)$.
There's a converse to this theorem, but a lone point is not a region, so I'm not sure if either of these theorems are relevant.
Is the only way to go back and find at which points
$lim_{h\to0} \frac{f(z_{0}+h)-f(z_{0})}{h}$
exists?
Thanks.