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Math Help - Complex Analysis - Question on differentiability of a complex function

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    Complex Analysis - Question on differentiability of a complex function

    Let f: \mathbb{C} \rightarrow \mathbb{C} be such that f(z)=Re(z). How would you go about determining where f is differentiable?
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    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
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    Re: Complex Analysis - Question on differentiability of a complex function

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

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

    So, for \rho\to 0 the limit that defines f'(z) depends on \theta .
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    Re: Complex Analysis - Question on differentiability of a complex function

    Thanks!
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