コーシー・リーマンの方程式
∂ ∂ z = 1 2 ( ∂ ∂ x − i ∂ ∂ y ) , ∂ ∂ z ̄ = 1 2 ( ∂ ∂ x + i ∂ ∂ y ) {\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}{\Bigr )},\;\;\;{\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}{\Bigr )}}
極限 Δ z → 0 , Δ z ̄ → 0 {\displaystyle \Delta z\rightarrow 0,\Delta {\bar {z}}\rightarrow 0} では上の等式は以下のように書くことができる。
d f d z | z = z 0 = ∂ f ∂ z | z = z 0 + ∂ f ∂ z ̄ | z = z 0 ⋅ d z ̄ d z + η ( Δ z ) , ( Δ z ≠ 0 ) . {\displaystyle \left.{\frac {df}{dz}}\right|_{z=z_{0}}=\left.{\frac {\partial f}{\partial z}}\right|_{z=z_{0}}+\left.{\frac {\partial f}{\partial {\bar {z}}}}\right|_{z=z_{0}}\cdot {\frac {d{\bar {z}}}{dz}}+\eta (\Delta z),\;\;\;\;(\Delta z\neq 0).