Question

Is f(z) = sin(z) an entire function? Why? What about cos(z)?

Answer 1

Question 1.

Is $f(z)=\sin(z)$ an entire function? Why? What about $\cos(z)$?

Solution. By definition,

$\sin z=\frac{e^{iz}-e^{-iz}}{2i},\ \ \cos z=\frac{e^{iz}+e^{-iz}}{2}.$

We know that the exponential function $g(z)=e^{z}$ and any polynomial are the entire functions. The class of entire functions is closed under the composition, so $\sin z$ and $\cos z$ are entire as the compositions of $e^{z}$ and linear functions. $\square$