5. What is the most general analytic function $f(x + iy)$ whose component functions are
polynomials of homogeneous degree 2.
That is, assuming $f(x+iy) = u(x, y) + iv(x, y)$ is analytic and $u(x, y) = ax^2 + bxy + cy^2$
and $v(x, y) = dx^2 + exy + fy^2$, what can we say about the relationship between the
coefficients $a$, $b$, $c$, $d$, $e$ and $f$? Recall, if $f$ is analytic, $u$ and $v$ must be harmonic.
