Question 3.31: Prove the equivalence of the operators: (a) ∂/∂x = ∂/∂z+∂/∂z...
Prove the equivalence of the operators:
(a) \frac{\partial}{\partial x}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}
(b) \frac{\partial}{\partial y}=i\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right) where z=x+i y, \bar{z}=x-i y.
If F is any continuously differentiable function, then
(a) \frac{\partial F}{\partial x}=\frac{\partial F}{\partial z} \frac{\partial z}{\partial x}+\frac{\partial F}{\partial \bar{z}} \frac{\partial \bar{z}}{\partial x}=\frac{\partial F}{\partial z}+\frac{\partial F}{\partial \bar{z}} showing the equivalence \frac{\partial}{\partial x}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}.
(b) \frac{\partial F}{\partial y}=\frac{\partial F}{\partial z} \frac{\partial z}{\partial y}+\frac{\partial F}{\partial \bar{z}} \frac{\partial \bar{z}}{\partial y}=\frac{\partial F}{\partial z}(i)+\frac{\partial F}{\partial \bar{z}}(-i)=i\left(\frac{\partial F}{\partial z}-\frac{\partial F}{\partial \bar{z}}\right) showing the equivalence \frac{\partial}{\partial y}=i\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right).