# Math Help - continous in the complex

1. ## continous in the complex

How do i prove sin z is continuous in the complex region?

2. If you mean using the definition of continuity, the same way you would in any region- show that, for any complex number, $z_0$, given any $\epsilon> 0$, there exist a real number $\delta> 0$ so that if $|z-z_0|< \delta$, then $|sin(z)- sin(z_0)|< \epsilon$.
Exactly how you would do that, depends on exactly how you are defining sin(z) for z complex. If you are defining sin(z) in terms of its Taylor series, just subtract the Taylor series for sin(z) and $sin(z_0)$. If you defining sin(z) as $\frac{e^{z)- e^{-z}}{2i}$, are you allowed to use the fact that $e^z$ is continuous?