$x = r \cos \theta, y = r \sin \theta$ より、 \begin{align} \frac{\partial x}{\partial r} = \cos \theta , \ \ \frac{\partial x}{\partial \theta} = -r \sin \theta , \ \ \frac{\partial y}{\partial r} = \sin \theta , \ \ \frac{\partial y}{\partial \theta} = r \cos \theta \end{align} なので、 \begin{align} \frac{\partial u}{\partial r} &= \frac{\partial x}{\partial r} \frac{\partial u}{\partial x} + \frac{\partial y}{\partial r} \frac{\partial u}{\partial y} \\ &= \cos \theta \frac{\partial u}{\partial x} + \sin \theta \frac{\partial u}{\partial y} , \\ \frac{\partial u}{\partial \theta} &= \frac{\partial x}{\partial \theta} \frac{\partial u}{\partial x} + \frac{\partial y}{\partial \theta} \frac{\partial u}{\partial y} \\ &= -r \sin \theta \frac{\partial u}{\partial x} + r \cos \theta \frac{\partial u}{\partial y} , \\ \frac{\partial v}{\partial r} &= \frac{\partial x}{\partial r} \frac{\partial u}{\partial x} + \frac{\partial y}{\partial r} \frac{\partial u}{\partial y} \\ &= \cos \theta \frac{\partial v}{\partial x} + \sin \theta \frac{\partial v}{\partial y} , \\ \frac{\partial v}{\partial \theta} &= \frac{\partial x}{\partial \theta} \frac{\partial u}{\partial x} + \frac{\partial y}{\partial \theta} \frac{\partial u}{\partial y} \\ &= -r \sin \theta \frac{\partial v}{\partial x} + r \cos \theta \frac{\partial v}{\partial y} \end{align} である。 さらに、コーシー・リーマンの方程式 \begin{align} \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} , \ \ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \end{align} を使うと、 \begin{align} \frac{\partial u}{\partial r} &= \cos \theta \frac{\partial u}{\partial x} + \sin \theta \frac{\partial u}{\partial y} \\ &= \cos \theta \frac{\partial v}{\partial y} - \sin \theta \frac{\partial v}{\partial x} \\ &= \frac{1}{r} \frac{\partial v}{\partial \theta} , \\ \frac{\partial v}{\partial r} &= \cos \theta \frac{\partial v}{\partial x} + \sin \theta \frac{\partial v}{\partial y} \\ &= - \cos \theta \frac{\partial u}{\partial y} + \sin \theta \frac{\partial u}{\partial x} \\ &= - \frac{1}{r} \frac{\partial u}{\partial \theta} \end{align} を得る。