Question:
(a) Show that $u(x,y)=-8x^{3}+24xy^{2}+4xy$ is a harmonic function.
(b) Find a harmonic conjugate function $v(x,y)$.
(c) Consider $u(x,y) + iv(x,y)$ and write it as a function $f(z)$ of $z=x+iy$.
Answer:
(a)
$
\begin{align*}
&u_x=-24x^{2}+24y^{2}+4y\\
&u_{xx}=-48x\\
&u_y=48xy+4x\\
&u_{yy}=48x\\
\Rightarrow &u_xx+u_yy=-48x+48x=0\\
\Rightarrow &\text{It's harmonic.}\square
\end{align*}
$
(b)
$
\begin{align*}
v_x&=-u_y=-48x-4x\\
v_y&=u_x=-24x^{2}+24y^{2}+4y\\
\Rightarrow & v(x,y)=\int{(-48x-4x)}dx+\phi(y)=-(24x^{2}y+2x^{2})+\phi(y)\\
v_y&=-24x^{2}+\phi'(y)\\
\Rightarrow &-24x^{2}+24y^{2}+4y=-24x^{2}+\phi'(y)\\
\Rightarrow &\phi(y)=\int{(24y^{2}+4y)}dy=8y^{3}+2y^{2}+C\\
\Rightarrow &v(x,y)=-24x^{2}y-2x^{2}+8y^{3}+2y^{2}+C\\
\end{align*}
$
(c)
$
\begin{align*}
u(x,y)& + iv(x,y)\\
&=-8x^{3}+24xy^{2}+4xy-24ix^{2}y-2ix^{2}+8iy^{3}+2iy{2}+iC\\
&=-8(x^{3}-iy^{3}-3xy^{2}+3ix^{2}y)-2i(x^{2}-y^{2}+2ixy)+iC\\
&=-8(x+iy)^{3}-2i(x+iy)^{2}+iC\\
&=-8z^{3}-2iz^{2}+iC\\
\Rightarrow & f(z)=-8z^{3}-2iz^{2}+iC
\end{align*}
$
