Find the imaginary part whose real part is u= x3

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written 4.5 years ago by

teamques10 ★ 70k

$\displaystyle With\ u=x^3-3xy^2+3x^2+1,\ \\[2ex] $

$\displaystyle we\ have\\[2ex] Ux=3x^2-3y^2+6x\ \ and\\[2ex] Uy=\ -6xy-6y. \\[2ex] $

$\displaystyle Therefore,\\[2ex] Uxx=^x+6\ and\\[2ex] Uy=-6x-6\ and\ so\\[2ex] Uxx+Uyy=0. \\[2ex] $

$\displaystyle Becasue\ U\ satisfies\ Laplace^{'}\ equation,\ there\ \ exists\ a\ conjugate\ function\ v\left(x,y\right)that \\[2ex] \displaystyle satisfies\ the\ CR\ equations:Ux=Vu,\ Vx=-Uy. \\[2ex] $

(\displaystyle To\ find\ …

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