Cauchy Riemann equations, do these satisfy it??

Question

I have this question and am unsure of my approach.

I have applied the Cauchy Riemann conditions to it:

and found that this condition is true. Is that sufficient and does it make sense?

Answer 1

$u,v$ satisfy Cauchy-Riemann equations if and only if $f=u+iv$ is holomorphic.

But if $f$ is holomorphic, then so is \begin{align} g=\exp(if^2)&=\exp\big(i(u+iv)^2\big)=\exp\big(i(u^2-v^2)-2uv\big)\\ &=\exp(-2uv)\big(\cos(u^2-v^2)+i\sin(u^2-v^2)\big), \end{align} which implies that $$ \exp(-2uv)\cos(u^2-v^2)\quad\text{and}\quad \exp(-2uv)\sin(u^2-v^2), $$ satisfy Cauchy-Riemann equations.