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QUESTION Find the real and imaginary parts $u(x,y)$ and $v(x,y)$
of the function $f(z)=z^3$. Show directly that
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\item[(i)]
$u$ and $v$ satisfy the Cauchy-Riemann equations and

\item[(b)]
$u$ and $v$ are harmonic functions.

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ANSWER $z^3=((x+iy)^3=x^3-3xy^2+i(3x^2y-y^3)$ so that
$u(x,y)=x^3-3xy^2, \ v(x,y)=3x^2y-y^3$\\ Now the ortial
derivatives are $u_x=3x^2-3y^2,\ v_y=3x^2-3y^2,\ u_y=-6xy,\
v_x=6xy$,\\
 so that
$u$ and $v$ satisfy the Cauchy-Riemann equations. Also
$u_{xx}+v_{yy}=6x-6x=0$ so $u$ is harmonic and similarly $v$ is
harmonic.


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