I want to check if there is (or not) an analytic function on $\mathbb{C}$ \ ${0}$ such that $Im(f)=78+x^2-5y^6- \frac{5y}{3(x^2+y^3)}$. What I thought of doing is first applying the Identity Theorem which guarantees the uniqueness of the analytic function (if it really exists), so I just set $x=z$ and let $y=0$. So I would get $f(z)=u(z,0)+iv(z,0)$.
If $f$ is analytic then by the Cauchy-Riemann equations, we get $v_x=2x=-u_y$ and $v_y=0=u_x$. So $u$ does not depend on $x$, which contradicts that $u_y=-2x$. So there is no such analytic function by the Identity Theorem.
Is it that easy or did I just do complete nonsense by first applying the Identity Theorem then going to the partial derivatives? Can I do that? Thx.