[Math] Domain of complex function.

Question

I'm totally new to complex analysis topic.

Consider the complex function,

$$ f(z) = \frac{\ln(z^2)}{exp(iz)+1}$$

What is the domain of analyticity of this function?

Also I am wondering how should I sketch it?

Answer 1

The domain of this function is the entire complex plane without the origin, because $\ln$ is not defined there, and every point where $e^{iz}=-1$, because this would cause division by $0$. Thus, the domain of $f$ is $\Bbb C\backslash(\{0\}\cup\{2\pi n+\pi:n\in\Bbb Z\}).$

Although difficult, as stated in Doug M's comment, it is possible to sketch complex functions, albeit in a roundabout way. To sketch it, you can sketch the level surfaces of $\text{Re}[f(z)]$, $\text{Im[f(z)]}$, and $|f(z)|.$ This is what wolfram alpha often does to demonstrate simply the behavior of complex functions, for example $\ln(z)$:

Of course, they have colors, but you'll be conveying similar information by drawing level curves.