A problem on Complex function

Question

Let $f(z)=(z^3+1)\sin z^2,$ for $z \in \mathbb{C}.$ Let $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$ and $u, v$ are real valued functions. Then which of the following are true?

1) $u:\mathbb{R}^2 \to \mathbb{R} $ is infinitely differentiable.

2) $u$ is continuous but need not be differentiable.

3) $u$ is bounded

4) $f$ can be represented by an absolutely convergent power series $\sum_{n=0}^{\infty} a_{n} z^n$ for all $z \in \mathbb{c}$.

I tried to find $u(x,y)$ using $\sin z² = \sin (x^2-y^2) \cosh (2xy)+i \big(\cos(x^2-y^2)\sinh(2xy)\big).$

But it didn't help me much.

Could anyone please suggest me how to proceed? Thank you

Answer 1

1. Since $u$ is the real part of an analytical function, it is infinitely differentiable.
2. This is false, given what I wrote above.
3. It is false: $(\forall n\in\mathbb N):u\left(\sqrt{\frac\pi2+2n\pi},0\right)=\sqrt{\frac\pi2+2n\pi}^3+1$.
4. Yes, since $f$ is an entire function. Every entire function has this property.