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In Stein and Shakarchi's book Complex Analysis, Proposition 2.3 states that if $f$ is holomorphic at $z_0$, then $$ f'(z_0)=\frac{\partial f}{\partial z}(z_0)=2\frac{\partial u}{\partial z}(z_0).$$ The first equality is easy to derive (and it is even done in the textbook), but I don't quite see how to establish the second equality. It apparently follows from the Cauchy-Riemann equations, but I haven't been able to make heads or tails of how exactly to verify the identity? Please help me!

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$2f'(z_0) = \frac{\partial f}{\partial x}(z_0) - i \frac{\partial f}{\partial y}(z_0)$. Since $ f = u + iv$, we have $\frac{\partial f}{\partial x} (z_0) = \frac{\partial u}{\partial x}(z_0) + i \frac{\partial v}{\partial x}(z_0)$, and similarly, $\frac{\partial f}{\partial y} (z_0) = \frac{\partial u}{\partial y}(z_0) + i \frac{\partial v}{\partial y}(z_0)$.

By C-R, $\frac{\partial v}{\partial x}(z_0) = - \frac{\partial u}{\partial y} (z_0)$ and $\frac{\partial v}{\partial y} (z_0) = \frac{\partial u}{\partial x} (z_0)$.

So, $2 f'(z_0) = (u_x - i u_y) - i(u_y + iu_x) = 2(u_x - iu_y) = 4 u_z(z_0)$.

Divide by 2.

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