# Complex Analysis - Question on differentiability of a complex function

• Oct 4th 2011, 04:46 AM
worc3247
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?
• Oct 4th 2011, 04:50 AM
Daniiel
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
• Oct 4th 2011, 05:06 AM
FernandoRevilla
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 $\displaystyle \theta$ .
• Oct 4th 2011, 05:30 AM
worc3247
Re: Complex Analysis - Question on differentiability of a complex function
Thanks!