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Scalar complex holomorphic function - same derivative as scalar real function?

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My question is about the derivative of holomorphic complex functions.

Assume there is a function $f(x) := \mathbb{R} \rightarrow \mathbb{R}$ , and a function $f(z) := \mathbb{C} \rightarrow \mathbb{C}$. Further assume that both functions have a "counterpart", both in the complex and real domain (e.g. $\sin(x)$, $e^x$, $|.|$ or polynomials).

When $f(z)$ is holomorphic on $\mathbb{C}$ can we then conclude that $f'(x) = f'(z)$?

From my understanding the second Wirtinger derivative must be zero for any holomorphic function $f(z)$.

$$\frac{\partial f(z)}{\partial z*} = \frac{1}{2}\left(\frac{\partial f(z)}{\partial a}+i\frac{\partial f(z)}{\partial b}\right) = 0$$

Which somehow leads me to believe that $\frac{\partial f(z_0)}{\partial z} = f'(x_0)$ assuming $x_0 = z_0$. However, as this is only an "Intuition" from my side, and I am by no means a professional Mathematician. Maybe one of you can construct a counterexample to my observation.


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