- Suppose U and \(V\) are open sets in the complex plane. Prove that if \(f : U \to V\) and \(g : V \to \mathbb { C }\) are two functions that are diferentiable (in the real sense, that is, as functions of the two real variables x and \(y )\) , and \(h = g \circ f\) , then
\[ \frac {\partial h}{\partial z} = \frac {\partial g}{\partial z} \frac {\partial f}{\partial z} + \frac {\partial g}{\partial \overline {{z}}} \frac {\partial \overline {{f}}}{\partial z} \]
and
\[ \frac {\partial h}{\partial \overline {{z}}} = \frac {\partial g}{\partial z} \frac {\partial f}{\partial \overline {{z}}} + \frac {\partial g}{\partial \overline {{z}}} \frac {\partial \overline {{f}}}{\partial \overline {{z}}}. \]
This is the complex version of the chain rule.
Solution
Write the real-differentiable map \(f\) locally as a function of \(z\) and \(\bar z\). Its differential is \[ df=f_z\,dz+f_{\bar z}\,d\bar z, \qquad d\bar f=\bar f_z\,dz+\bar f_{\bar z}\,d\bar z. \] Likewise, for \(g\), viewed at the point \(f(z)\), \[ dg=g_z\,dw+g_{\bar z}\,d\bar w. \] For \(h=g\circ f\), substitute \(w=f(z)\): \[ \begin{aligned} dh &=g_z\,df+g_{\bar z}\,d\bar f\\ &=g_z(f_z\,dz+f_{\bar z}\,d\bar z) +g_{\bar z}(\bar f_z\,dz+\bar f_{\bar z}\,d\bar z)\\ &=(g_zf_z+g_{\bar z}\bar f_z)\,dz +(g_zf_{\bar z}+g_{\bar z}\bar f_{\bar z})\,d\bar z. \end{aligned} \] Comparing the coefficients of \(dz\) and \(d\bar z\) yields \[ h_z=g_zf_z+g_{\bar z}\bar f_z, \qquad h_{\bar z}=g_zf_{\bar z}+g_{\bar z}\bar f_{\bar z}, \] which are the desired formulas.