Every simply connected Riemann surface is conformally equivalent to exactly one of: the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, the complex plane $\mathbb{C}$, or the open unit disk $\mathbb{D} = \{z : |z| < 1\}$.

Let $\Omega \subset \mathbb{C}$ be a connected open set, and let $f, g: \Omega \to \mathbb{C}$ be holomorphic. Let $R: \mathbb{C}^2 \to \mathbb{C}$ be a holomorphic function of two complex variables $(w_1, w_2)$, so that $z \mapsto R(f(z), g(z))$ is holomorphic on $\Omega$. Suppose $R(f(z), g(z)) = 0$ for all $z$ in a set $S \subset \Omega$ with a limit point in $\Omega$. Then $R(f(z), g(z)) = 0$ for all $z \in \Omega$.

The Riemann zeta function $\zeta(s)$, initially defined for $\operatorname{Re}(s) > 1$, extends to a meromorphic function on all of $\mathbb{C}$. The only singularity is a simple pole at $s = 1$ with residue $1$. Moreover, $\zeta$ satisfies the **functional equation**: \begin{align*} \pi^{-s/2}\,\Gamma\!\left(\frac{s}{2}\right)\zeta(s) &= \pi^{-(1-s)/2}\,\Gamma\!\left(\frac{1-s}{2}\right)\zeta(1-s). \end{align*}

For $\operatorname{Re}(s) > 1$, the Riemann zeta function satisfies \begin{align*} \Gamma(s)\,\zeta(s) &= \int_0^\infty \frac{t^{s-1}}{e^t - 1} \, dt. \end{align*}

Let $\sum_{n=0}^\infty a_n z^{\lambda_n}$ be a power series with radius of convergence $R \in (0, \infty)$, where the exponents $\lambda_n$ are positive integers satisfying the gap condition \begin{align*} \liminf_{n \to \infty} \frac{\lambda_{n+1}}{\lambda_n} &> 1. \end{align*} Then the circle $|z| = R$ is a natural boundary of $\sum_{n=0}^\infty a_n z^{\lambda_n}$: the function cannot be analytically continued across any arc of this circle.

Let $\Omega \subset \mathbb{C}$ be a simply connected domain, and let $f$ be a holomorphic function element defined near a point $a \in \Omega$. Suppose $f$ can be analytically continued along every path in $\Omega$ starting from $a$. Then these continuations define a single-valued holomorphic function on all of $\Omega$.

Let $\Omega^+$ be a connected open subset of the upper half-plane $\{\operatorname{Im}(z) > 0\}$, and suppose the boundary $\partial \Omega^+$ contains an open segment $I \subset \mathbb{R}$ of the real axis. Let $\Omega^-$ denote the reflection of $\Omega^+$ across the real axis: $\Omega^- := \{\bar{z} : z \in \Omega^+\}$. Suppose $f: \Omega^+ \cup I \to \mathbb{C}$ is holomorphic on $\Omega^+$, continuous on $\Omega^+ \cup I$, and real-valued on $I$. Then $f$ extends to a holomorphic function on $\Omega^+ \cup I \cup \Omega^-$ via \begin{align*} F(z) &= \begin{cases} f(z) & z \in \Omega^+ \cup I \\ \overline{f(\bar{z})} & z \in \Omega^-. \end{cases} \end{align*}

A function $f: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ is meromorphic on $\widehat{\mathbb{C}}$ if and only if $f$ is a rational function, i.e., $f = P/Q$ for polynomials $P, Q \in \mathbb{C}[z]$ with $Q \not\equiv 0$.

Let $\Omega \subset \mathbb{C}$ be open and $f \in \mathcal{M}(\Omega)$, $f \not\equiv 0$. Then $f'/f \in \mathcal{M}(\Omega)$. At a zero $z_0$ of $f$ of order $m$, the function $f'/f$ has a simple pole with residue $+m$. At a pole $z_0$ of $f$ of order $m$, the function $f'/f$ has a simple pole with residue $-m$.

Let $(z_j)_{j \in \mathbb{N}} \subset \mathbb{C}$ be a sequence with $|z_j| \to \infty$, and for each $j$, let $P_j(1/(z - z_j))$ be a polynomial in $1/(z - z_j)$ with no constant term (the prescribed principal part at $z_j$). Then there exists a meromorphic function $f: \mathbb{C} \to \hat{\mathbb{C}}$ with poles exactly at the $z_j$ and with principal part $P_j(1/(z-z_j))$ at each $z_j$. Moreover, any two such functions $f$ and $\tilde{f}$ differ by an entire function: $f - \tilde{f}$ is holomorphic on all of $\mathbb{C}$.