Question

If $\varphi (x,y)$ and $\Psi (x,y)$ are functions with continuous second derivatives, then $\varphi (x,y)+i\Psi (x,y)$ can be expressed as an analytic function of $x+iy(i=\sqrt{-1})$ when

• A. $\frac{\partial \varphi }{\partial x}=\frac{\partial \Psi }{\partial x},\frac{\partial \varphi }{\partial y}=\frac{\partial \Psi }{\partial y}$
• B. $\frac{\partial \varphi }{\partial y}=-\frac{\partial \Psi }{\partial x},\frac{\partial \varphi }{\partial x}=\frac{\partial \Psi }{\partial y}$
• C. $\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\Psi }{\partial y^2}=\frac{{\partial }^2\Psi }{\partial x^2}+\frac{{\partial }^2\Psi }{\partial y^2}=1$
• D. $\frac{\partial \varphi }{\partial x}+\frac{\partial \Psi }{\partial y}=\frac{\partial \Psi }{\partial x}+\frac{\partial \Psi }{\partial y}=0$

Answer 1

The correct option is B $\frac{\partial \varphi }{\partial y}=-\frac{\partial \Psi }{\partial x},\frac{\partial \varphi }{\partial x}=\frac{\partial \Psi }{\partial y}$

Let $w=\varphi (x,y)+i\Psi (x,y)$, then $w$ is analytic when
$\frac{\partial \varphi }{\partial x}=\frac{\partial \Psi }{\partial y},\frac{\partial \varphi }{\partial y}=\frac{-\partial \Psi }{\partial x}$ (C-R equations)