ウィルティンガーの微分
補題3.2 g ∈ C 1 ( Ω ′ , Ω ) {\displaystyle g\in C^{1}(\Omega ^{\prime },\Omega )} および f ∈ C 1 ( Ω , Ω ′ ′ ) {\displaystyle \scriptstyle f\in C^{1}(\Omega ,\Omega ^{\prime \prime })} であれば、 i = 1 , ... , m {\displaystyle i=1,\dots ,m} に対し、以下の形のチェインルールが成り立つ
∂ ∂ z i ( f ∘ g ) = ∑ j = 1 n ( ∂ f ∂ z j ∘ g ) ∂ g j ∂ z i + ∑ j = 1 n ( ∂ f ∂ z ̄ j ∘ g ) ∂ g ̄ j ∂ z i {\displaystyle {\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}}