Is this f(z) function analytic? Is $f(z) = z^2\sin z$  an analytic function for $z \in \mathbb{C}$ ?
$z = x + iy $
I really don't know how to split this at this format $f(x,y) = P(x,y) + iQ(x,y)$, so I can prove that if this function is analytic with neccesary Cauchy–Riemann equations. Which are $ P_x = Q_y$ and $P_y = -Q_x$.
I can't get out of this $f(x+iy) = (x+iy)^2\sin(x+iy)$. How can I get rid of $i$ in $\sin(x+iy)$?

Using Cauchy-Riemann here is unnecessary work. It's easier (even if you haven't done so before) to show that $\sin z$ is holomorphic (analytic), and that the product of two holomorphic functions is again holomorphic.
