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Let $f : \mathbb C^n \rightarrow \mathbb C^m$ be holomorphic and $g : \mathbb C^m \rightarrow \mathbb C$ be smooth. I am looking for a simple formula for the mixed partials $\partial_i \partial_{\bar j}(g\circ f)$, where $\partial_i$ and $\partial_{\bar j}$ denote complex differentials,

$$\partial_i u = \frac{\partial u}{\partial x_i} - \sqrt{-1} \frac{\partial u}{\partial y_i}$$

and

$$\partial_{\bar j} u = \frac{\partial u}{\partial x_j} + \sqrt{-1} \frac{\partial u}{\partial y_j}.$$

Applying the chain rule seems to give me a rather nasty looking expression involving summation over several different indices. Is there a simpler way to write this?

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1 Answer 1

up vote 2 down vote accepted

Firstly, I think the standard notations of $\partial_i$ and $\partial_{\bar{j}}$ differ from yours with a factor $\frac{1}{2}$. See, for example, here. Nevertheless, the following calculations are independent of the choice of notations.

Since $f$ is holomorphic, $\partial_{\bar{i}}f_k=\partial_{\bar{i}}\partial_jf_k=0$, $\forall i,j,k$. Then $$\partial_{\bar{j}}(g\circ f)=\sum_{k=1}^m\partial_{\bar{k}}g\circ f\cdot \overline{\partial_j f_k},$$

and

$$\partial_i\partial_{\bar{j}}(g\circ f)=\sum_{k,l=1}^m\partial_l\partial_{\bar{k}}g\circ f\cdot\partial_if_l \cdot\overline{\partial_j f_k}.$$

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