Question

Using the Cauchy –Riemann equations verify the following is analytic or not

i) 𝑥^2 − 𝑦^2 + 2𝑖𝑥𝑦

ii) 𝑥^2 + 𝑦^2 − 2𝑖𝑥𝑦

Answer 1

$i:\\u\left( x,y \right) =x^2-y^2\\v\left( x,y \right) =2xy\\\frac{\partial u}{\partial x}=2x,\frac{\partial u}{\partial y}=-2y\\\frac{\partial v}{\partial x}=2y,\frac{\partial v}{\partial y}=2x\\\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\Rightarrow f\,\,is\,\,analytic\\ii:\\u\left( x,y \right) =x^2+y^2\\v\left( x,y \right) =-2xy\\\frac{\partial u}{\partial x}=2x\\\frac{\partial v}{\partial y}=-2x\\\frac{\partial u}{\partial x}\ne \frac{\partial v}{\partial y}\Rightarrow f\,\,is\,\,not\,\,analytic$