Finding a Real valued Function to Create Holomorphism

Question

I am asked whether it is possible to find a real function $v$ such that $$x^3+y^3+iv$$ is holomorphic.

Should I basically be working backwards from the Cauchy-Riemann equations? That makes logical sense to me, but I don't really see how it would come together exactly.

Thanks

Answer 1

If $x^3 + y^3 + iv$ were holomorphic, then the real part $u(x, y) = x^3 + y^3$ would be harmonic, that is, $u$ would satisfy $\nabla^2 u = 0$. But $\nabla^2 u = 6(x + y) \ne 0$, so $u$ is not harmonic, so $u + iv = x^3 + y^3 + iv$ cannot be holomorphic for any $v$.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!