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Given a holomorphic function $f(z)$, we define the derivative $f'(z)$ as $$ \lim_{z\to z_0} \frac{f(z) - f(z_0)}{z-z_0} $$ Using this definition, you can prove the Cauchy-Riemann equations by analyzing the derivative limit from different paths. Doing this,and writing $f$ as $f = u + iv$ you get $$ f'(z) = u_x + iv_x = v_y - i u_y $$ My question is, if we define the $n$-th derivative as $f^{(n)} = \left(f^{(n-1)}\right)'$, and we know that the $n$-derivative exists, is there a way to obtain a formula for the $n$-th derivative of $f$ in terms of the partial derivatives of $u$ and $v$?

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You can apply the same argument to $f'= U+iV$: $$ f'' = U_x + iV_x = u_{xx} + i v_{xx} \, . $$ The generalization to higher order derivatives should be obvious now.

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