Real part of $\sinh(z)$

Question

Is it true that for every $z \in \mathbb{C}$

$$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z \end{align}$$

Answer 1

This is true because the coefficients of the power series expansion of these entire functions are real.