Complex Analysis - Question on differentiability of a complex function

Let $\displaystyle f: \mathbb{C} \rightarrow \mathbb{C}$ be such that f(z)=Re(z). How would you go about determining where f is differentiable?

Re: Complex Analysis - Question on differentiability of a complex function

Let z = x + iy, and f(z) = u + iv,

when you do that you will find what u and v are, then you can use the C-R equations and find if/where they hold

Re: Complex Analysis - Question on differentiability of a complex function

Another way: if $\displaystyle h=\rho e^{i\theta}$ then,

$\displaystyle \frac{f(z+h)-f(z)}{h}=\frac{\textrm{Re}\;h}{h}=\ldots=\cos^2 \theta-i\sin \theta \cos \theta$

So, for $\displaystyle \rho\to 0$ the limit that defines $\displaystyle f'(z)$ depends on $\displaystyle \theta$ .

Re: Complex Analysis - Question on differentiability of a complex function