# Is there a formula for the $n$-th derivative of holomorphic function in terms of its partial derivatives?

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 $$v$$?

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.