Answer to Question #313819 in Complex Analysis for Jyo

Question #313819

Using the Cauchy โ€“Riemann equations verify the following is analytic or not


i) ๐‘ฅ^2 โˆ’ ๐‘ฆ^2 + 2๐‘–๐‘ฅ๐‘ฆ


ii) ๐‘ฅ^2 + ๐‘ฆ^2 โˆ’ 2๐‘–๐‘ฅ๐‘ฆ

1
Expert's answer
2022-03-19T02:40:17-0400

i:u(x,y)=x2โˆ’y2v(x,y)=2xyโˆ‚uโˆ‚x=2x,โˆ‚uโˆ‚y=โˆ’2yโˆ‚vโˆ‚x=2y,โˆ‚vโˆ‚y=2xโˆ‚uโˆ‚x=โˆ‚vโˆ‚y,โˆ‚uโˆ‚y=โˆ’โˆ‚vโˆ‚xโ‡’fโ€‰โ€‰isโ€‰โ€‰analyticii:u(x,y)=x2+y2v(x,y)=โˆ’2xyโˆ‚uโˆ‚x=2xโˆ‚vโˆ‚y=โˆ’2xโˆ‚uโˆ‚xโ‰ โˆ‚vโˆ‚yโ‡’fโ€‰โ€‰isโ€‰โ€‰notโ€‰โ€‰analytici:\\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


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