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QUESTION Show that $u(x,y)=2x-x^3+3xy^2$ is harmonic. Find a
conjugate harmonic function $v(x,y)$ and identify the
corresponding analytic function $u+iv$.

ANSWER  Easy to show $u_{xx}+u_{yy}=0$, so $u$ is harmonic. Let
$v$ be the conjugate harmonic function. Then
$$v_y=u_x=2-3x^2+3y^2$$ Thus $$v(x,y)=2y-3x^2y+y^3+\phi(x).$$ Now
$v_x=-6xy+\phi'(x)=-u_y=-6xy$. Thus $\phi(x)=$constant, so
$$v(x,y)=2y-3x^2y+y^3+\textrm{constant}$$ and $u+iv=2z-z^3+ic$,
where $c$ is a real constant. (Note that we have expressed this in
terms of $z=x+iy$.)

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