I was solving a problem yesterday and it has bugged me for the whole night, im not sure whether if I got it correct or not.
First I was asked if $f(x+iy)=x^2-y^2 + i\sqrt{|xy|}$ satisfies the C-R equations at $0$. So I found $\frac{\partial u}{\partial x}$ $=$ $\frac{\partial v}{\partial y}$ & $-\frac{\partial u}{\partial y}$ $=$ $\frac{\partial v}{\partial x}$
$u=x^2-y^2$ and $v=\sqrt{|xy|}$
So I found $\frac{\partial u}{\partial x} = 2x$, $-\frac{\partial u}{\partial y}=2y$, $\frac{\partial v}{\partial x}=\frac{1}{2}\frac{\sqrt{y}}{\sqrt{x}}$ and $\frac{\partial v}{\partial y}=\frac{1}{2}\frac{\sqrt{x}}{\sqrt{y}}$.
So obviously it does not satisfy the C-R equations.
I was wondering if I have to do anything else because it asks if it satisfies the C-R equations at $0$?
The next part of the question asks me if $f$ is differentiable at $0$. And it hints that I should consider $\mathrm{lim}_{r\rightarrow 0} \frac{f(re^{i\theta})}{re^{i\theta}}$. I assume they are meaning $f$ is complex differentiable? (not real differentiable?). How would I determine if f is differentiable at $0$?
Because it says on wikipedia that the sole existence of partial derivatives satisfying the Cauchy-Riemann equations is not enough to ensure complex differentiability at that point. It is necessary to make sure that u and v are real differentiable, which is a stronger condition than the existence of the partial derivatives but it is not necessary to require continuity of these partial derivatives. Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann equations at that point.
- The question I have is, since $f$ does not satisfy the C-R equations (see my calculations above), is there any need to do anything further (in question 2)? Can I just say its not differentiable at $0$? Why did they ask me to consider that limit?
Thanks alot, Im really stuck and slightly confused with all this..