Show that (a) [latex]\nabla \equiv \frac{\partial}{\partial x}+i \frac{\partial}{\partial y}=2 \frac{\partial}{\partial \vec{z}}[/latex], (b)[latex]\bar{\nabla} \equiv \frac{\partial}{\partial x}-i \frac{\partial}{\partial y}=2 \frac{\partial}{\partial z} [/latex] .

From the equivalences established in Problem 3.31 , we have (a) [latex]\nabla \equiv \frac{\partial}{\partial x}+i \frac{\partial}{\partial y}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}+i^2\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right)=2 \frac{\partial}{\partial \bar{z}}[/latex] (b) [latex]\bar{\nabla} \equiv \frac{\partial}{\partial x}-i \frac{\partial}{\partial y}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}-i^2\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right)=2 \frac{\partial}{\partial z}[/latex]

Question 3.32: Show that (a) ∇ ≡ ∂/∂x + i ∂/∂y = 2 ∂/∂z , (b) ∇ ≡ ∂/∂x - i ...

Schaum’s Outline of Complex Variables

Show that (a) $\nabla \equiv \frac{\partial}{\partial x}+i \frac{\partial}{\partial y}=2 \frac{\partial}{\partial \vec{z}}$ , (b) $\bar{\nabla} \equiv \frac{\partial}{\partial x}-i \frac{\partial}{\partial y}=2 \frac{\partial}{\partial z}$ .

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From the equivalences established in Problem 3.31, we have

(a) $\nabla \equiv \frac{\partial}{\partial x}+i \frac{\partial}{\partial y}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}+i^2\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right)=2 \frac{\partial}{\partial \bar{z}}$
(b) $\bar{\nabla} \equiv \frac{\partial}{\partial x}-i \frac{\partial}{\partial y}=\frac{\partial}{\partial z}+\frac{\partial}{\partial \bar{z}}-i^2\left(\frac{\partial}{\partial z}-\frac{\partial}{\partial \bar{z}}\right)=2 \frac{\partial}{\partial z}$