This particular question was asked in my complex analysis and I was not able to solve it. SO, I am asking for help.
Let p(z) , q(z) be two non-zero complex polynomials. Then p(z)$\bar{q(z)}$ is analytic iff
(i) p(z) is constant
(ii)p(z) q(z) is constant
(iii)q(z) is constant
(iv) $\bar{p(z)} $q(z) is constant
I think q(z) is constant should be answer as then p(z)$\bar{q(z)}$ will be analytic but answer is (iv) for which I have no explanation or which result should be used.
Can you please shed some light on how to prove it?
Thank you!!
Best Answer
The correct answer is (iii) as you you guessed. If $p(z)\overline {q(z)}$ is analytic then $\overline {q(z)}$ is anayltic on the region obtained by removing the zeros of $p$. But this implies that $q$ is a constant there. By continuity it is a constant on $\mathbb C$.