Recall that we can have unbounded functions with bounded area.
Examples:
Definition. A zero of a function is a value of z such that f(z) = 0.
For example, the zeros of \sin are n\pi + 0i for n \in \mathbb Z. This can be derived from the \sin(x+iy) = \sin x \cosh y + i \cos x \sinh y equation. Similarly, the zeros of \cos are (n+1/2)\pi. The zeros of \sinh and \cosh are n\pi i and (n+1/2)\pi i, respectively.
If w = \arcsin z, then z = \sin w and \begin{aligned} z &= \sin w \\ &= \frac {e^{iw}-e^{-iw}}{2i} \frac{e^{iw}}{e^{iw}}\\ &= \frac{e^{2iw}-1}{2ie^{iw}}\\ \implies 2ie^{ze^{iw}} &= e^{2iw}-1\\ \implies (e^{iw})^2 - 2iz(e^{iw}) - 1& = 0 \end{aligned} We can solve this quadratic using the complex quadratic formula, which doesn’t use \pm but instead uses (\cdot)^{1/2} as a multi-valued square root. So, \begin{aligned} \implies e^{iw} &= \frac{2iz + (-4z^2 + 4)^{1/2}}{2} \\ &= iz + (1-z^2)^{1/2}\\ \implies iw &= \log(iz + (1-z^2)^{1/2})\\ w =\arcsin z&= -i\log(iz + (1-z^2)^{1/2}) \end{aligned} Note that we have a multi-valued logarithm and for each of those, a double-valued square root. This makes it a lot more fun than real numbers.
Example: \arcsin (-i) = -i\log(1+z^{1/2})=-i\log(1\pm\sqrt 2). So we need to consider two logarithms. \log(1+\sqrt 2) = \ln (1+ \sqrt 2) + 2n\pi i is relatively fine. Then, \begin{aligned} \log (1-\sqrt 2) &= \ln|1-\sqrt 2| + \arg(1-\sqrt 2)\\ &=\ln (\sqrt 2 - 1) + (2n+1)\pi i \end{aligned} Putting these together, we get \arcsin (-i) is -i(\ln(1+\sqrt 2)+2n\pi i) and -i(\ln(\sqrt 2-1)+(2m+1)\pi i) for n, m \in \mathbb Z.
Topology is the study of topos, space. Our basic building block is some ball around an arbitrary point in \mathbb C.
Definition. Given z_0 \in \mathbb C and \epsilon > 0, B_\epsilon(z_0) denotes the (open) ball of radius \epsilon about z_0, a.k.a. an \epsilon-neighbourhood of z_0. In set notation, B_\epsilon(z_0) = \{z : |z-z_0| < \epsilon\}. Similarly, \overline B_\epsilon(z_0) is the closed ball of radius \epsilon about z_0 (a closed \epsilon-neighbourhood of z_0) given by \{ z : |z-z_0| \le \epsilon\}. A deleted \epsilon-neighbourhood of z_0 is \{z : 0 < |z-z_0| < \epsilon\}.
Note that the only feature of \mathbb C used by this definition is |\cdot|, the modulus. That is, \begin{aligned} |z-z_0| &= \sqrt{(x-x_0)^2 + (y-y_0)^2} \\ &= \|(x,y)-(x_0,y_0)\|_{\mathbb R^2} \\ &= d((x,y), (x_0,y_0))_\mathbb R \\ &= d(z, z_0)_\mathbb C \end{aligned} This has obvious analogues to \mathbb R with d(x,y)_\mathbb R = |x-y| being the absolute value distance. Balls in \mathbb R are just intervals.
Definition. Given \Omega \subseteq \mathbb C, z \in \mathbb C is an interior point of \Omega if there exists \epsilon > 0 such that B_\epsilon(z) \subset \Omega. Note that this implies B_{\epsilon'}(z) \subset \Omega for all 0<\epsilon' < \epsilon.
Definition. z \in \mathbb C is an exterior point of \Omega if there exists \epsilon > 0 such that B_\epsilon(z) \cap \Omega = \emptyset.
Definition. z \in \mathbb C is a boundary point of \Omega if for all \epsilon > 0, B_\epsilon(z) \cap \Omega \ne \emptyset and B_\epsilon(z) \cap \Omega^c \ne \emptyset. That is, any \epsilon-neighbourhood around z contains points inside and outside \Omega. Here, $^c $ denotes the complement, that is \mathbb C \setminus \Omega.