Identify $\mathbb{C}$ with $\mathbb{R}^2$ by sending $x+iy$ to $(x,y)$. Let \begin{align*} A: \mathbb{R}^2 \to \mathbb{R}^2 \end{align*} be a real [linear map](/page/Linear%20Map) whose matrix with respect to the standard basis of $\mathbb{R}^2$ is \begin{align*} \begin{pmatrix} a & c \\ b & d \end{pmatrix}, \end{align*} where $a,b,c,d \in \mathbb{R}$. Then $A$ is complex-linear if and only if $a=d$ and $c=-b$. In that case, under the identification $\mathbb{C}\cong\mathbb{R}^2$, the map $A$ is multiplication by the complex number $a+ib$.

Let $\Omega \subset \mathbb C$ be open, let \begin{align*} f: \Omega &\to \mathbb C \end{align*} be a function whose real and imaginary parts \begin{align*} u: \Omega &\to \mathbb R, & v: \Omega &\to \mathbb R \end{align*} have first partial derivatives at $z_0 = x_0 + i y_0 \in \Omega$, and write $f = u + iv$. Define the Wirtinger derivatives at $z_0$ by \begin{align*} \partial_z f(z_0) &= \frac{1}{2}\left(\partial_x f(z_0) - i\partial_y f(z_0)\right), \\ \partial_{\bar z} f(z_0) &= \frac{1}{2}\left(\partial_x f(z_0) + i\partial_y f(z_0)\right). \end{align*} Then $f$ satisfies the [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) at $z_0$ if and only if \begin{align*} \partial_{\bar z} f(z_0)=0. \end{align*} When these equations hold at $z_0$, one has \begin{align*} \partial_z f(z_0)=\partial_x u(z_0)+i\partial_x v(z_0). \end{align*}…

Let $\Omega \subset \mathbb C$ be open, let $u \in C^2(\Omega;\mathbb R)$ satisfy $\Delta u = 0$ on $\Omega$, and let $z_0 \in \Omega$. Then there exist an open neighbourhood $U \subset \Omega$ of $z_0$ and a function $v: U \to \mathbb R$ such that the function $F: U \to \mathbb C$ defined by $F(z)=u(z)+iv(z)$ is holomorphic on $U$.

Let $F: \mathbb{R} \to [0,1]$ be a function. There exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a real-valued [random variable](/page/Random%20Variable) $X:(\Omega,\mathcal{F}) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ such that \begin{align*} F(x) = \mathbb{P}(X \leq x) \end{align*} for every $x \in \mathbb{R}$ if and only if $F$ is nondecreasing, right-continuous, and satisfies \begin{align*} \lim_{x \to -\infty} F(x) &= 0, & \lim_{x \to \infty} F(x) &= 1. \end{align*}

Let $\mathbb N:=\{1,2,3,\ldots\}$. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $\mathcal B(\mathbb R)$ be the Borel $\sigma$-algebra on $\mathbb R$, and let $X_n:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ for $n\in\mathbb N$ and $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be real-valued random variables. Then the following implications hold: if $X_n\xrightarrow{a.s.}X$, then $X_n\xrightarrow{\mathbb P}X$; if $X_n\xrightarrow{L^p}X$ for some real number $p\ge1$, then $X_n\xrightarrow{\mathbb P}X$; and if $X_n\xrightarrow{\mathbb P}X$, then $X_n\xrightarrow{d}X$.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be a real-valued random variable. Let $\mathcal L^1$ denote one-dimensional Lebesgue measure on $(\mathbb R,\mathcal B(\mathbb R))$. Then $X$ is absolutely continuous, meaning that its law $\mu_X:=\mathbb P\circ X^{-1}$ satisfies $\mu_X\ll\mathcal L^1$, if and only if there exists a nonnegative Borel measurable function $f_X:\mathbb R\to[0,\infty]$ such that \begin{align*} \mathbb P(X\in A)=\int_A f_X(x)\,d\mathcal L^1(x) \end{align*} for every $A\in\mathcal B(\mathbb R)$. The function $f_X$ is unique up to equality $\mathcal L^1$-a.e.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$, and let $X:(\Omega,\mathcal F)\to([0,\infty),\mathcal B(\mathbb R)|_{[0,\infty)})$ be a nonnegative real-valued random variable. Then, as an identity in $[0,\infty]$, \begin{align*} \mathbb E[X]=\int_0^\infty \mathbb P(\{\omega\in\Omega:X(\omega)>t\})\,d\mathcal L^1(t). \end{align*}

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$, and let \begin{align*} X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R)) \end{align*} be a real-valued random variable. Define the distribution function \begin{align*} F_X:\mathbb R&\to[0,1]\\ x&\mapsto \mathbb P(\{\omega\in\Omega:X(\omega)\le x\}). \end{align*} Then for all real numbers $a<b$, \begin{align*} \mathbb P(\{\omega\in\Omega:a<X(\omega)\le b\})=F_X(b)-F_X(a). \end{align*} Moreover, for every real number $a$, \begin{align*} \mathbb P(\{\omega\in\Omega:X(\omega)=a\})=F_X(a)-\lim_{x\uparrow a}F_X(x), \end{align*} where $\lim_{x\uparrow a}F_X(x)$ denotes the left-hand limit of $F_X$ at $a$.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $(E,\mathcal E)$ be a measurable space, and let $X:(\Omega,\mathcal F)\to(E,\mathcal E)$ be a [random variable](/page/Random%20Variable), meaning an $\mathcal F/\mathcal E$-measurable map. Define the distribution, or pushforward measure, $\mu_X:\mathcal E\to[0,1]$ by \begin{align*} \mu_X(A)=\mathbb P(X^{-1}(A)) \end{align*} for every $A\in\mathcal E$, where $X^{-1}(A)=\{\omega\in\Omega:X(\omega)\in A\}$. Then for every nonnegative $\mathcal E$-measurable function $g:E\to[0,\infty]$, \begin{align*} \int_\Omega g(X(\omega))\,d\mathbb P(\omega)=\int_E g(x)\,d\mu_X(x). \end{align*}

Let $G$ and $H$ be groups written multiplicatively, let $S \subset G$, and suppose $G = \langle S\rangle$, where $\langle S\rangle$ denotes the subgroup of $G$ generated by $S$. If $\varphi,\psi: G \to H$ are group homomorphisms such that $\varphi(s)=\psi(s)$ for every $s \in S$, then $\varphi=\psi$ as functions $G \to H$.