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Let $n \ge 2$ be an integer and let $\zeta$ be a primitive root of unity. Then $$\sum_{k=0}^{n-1} \zeta^k = 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1} = 0.$$

A complex logarithm of a nonzero complex number $z$ is a complex number $w$ with $e^w = z$, see the definition. Each $z \neq 0$ has infinitely many logarithms, which differ by integer multiples of $2\pi i$. A complex logarithm function is a continuous choice of a logarithm on an open subset of $\mathbb{C} \setminus \{0\}$.

The line integral of a scalar field along a curve adds up the values of the field along the curve. Each value is weighted by a small length of the curve. For a non-negative field on a plane curve, it is the area of the curtain between the curve and the graph of the field.

The multiplicative order of an integer $a$ modulo $n$, with $a$ coprime to $n$, is the smallest positive exponent $k$ such that $a^k \equiv 1 \pmod n$. The definition of the multiplicative order gives the precise form.

A topological property is a property of topological spaces such that, if a space has it, each homeomorphic space also has it. Examples are compactness, connectedness, separability, metrizability and the Hausdorff property.

Reduction of order is a method to find a second solution of a homogeneous linear equation from a known one. Let $p, q$ be continuous on an open interval $I$, and let $y_1$ be a solution of $y'' + p(x) y' + q(x) y = 0$ with no zero on $I$. Put $y = v\, y_1$. Then $y$ is a solution if and only if $y_1 v'' + (2 y_1' + p\, y_1) v' = 0$, a first-order linear equation for $v'$. One solution gives $y_2 = y_1 \int \frac{e^{-\int p\, dx}}{y_1^2}\, dx$, and $(y_1, y_2)$ is a fundamental system on $I$.

Let $a < b$ and let $f, g \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Then there is $c \in (a, b)$ with $$(f(b) - f(a)) \, g'(c) = (g(b) - g(a)) \, f'(c).$$

Let $X$ be a topological space and let $S \subseteq X$. Then the closure $\overline{S}$ is a closed set of $X$.

Let $(a_n)$ be a decreasing real sequence: $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$. Assume that $(a_n)$ is bounded below. Then $(a_n)$ converges, and its limit is the infimum $\inf \{a_n : n \in \mathbb{N}\}$.

Let $U$ be a nonempty open subset of $\mathbb{C}$ with $U \neq \mathbb{C}$. Suppose that $U$ is simply connected: $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path. Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$. Then there is a biholomorphic map $f$ from $U$ onto $\mathbb{D}$, that is, a bijective holomorphic map with a holomorphic inverse. Moreover, for each $z_0 \in U$ there is exactly one such $f$ with $f(z_0) = 0$ and $f'(z_0)$ a positive real number.

A branch of the logarithm on an open set $U \subseteq \mathbb{C} \setminus \{0\}$ is a continuous function $L$ on $U$ with $e^{L(z)} = z$, see the definition. On a domain, two branches differ by a constant integer multiple of $2\pi i$.

The centraliser of a subset $S$ of a group is the set of elements that commute with each element of $S$. It is a subgroup.

A one-form is a differential form of degree one. At each point it gives a linear function of tangent vectors. One-forms are the natural integrands of line integrals, and the differential of a function is the basic example.

The complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ are not planar.

Partial fraction decomposition writes a rational function as a sum of a polynomial and simple fractions. Let $K$ be a field, and let $P/Q$ be a rational function with $P, Q \in K[X]$ and $Q \neq 0$. Assume that $Q = \prod_{i=1}^{r} q_i^{m_i}$, where the $q_i$ are distinct monic irreducible polynomials. Then there are unique polynomials $B$ and $A_{ij}$, with $\deg A_{ij} < \deg q_i$, such that $$\frac{P}{Q} = B + \sum_{i=1}^{r} \sum_{j=1}^{m_i} \frac{A_{ij}}{q_i^{\,j}}.$$

A commutative ring is a ring in which the multiplication is commutative. Commutative algebra is the study of these rings. The definition of a commutative ring gives the precise condition.

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be holomorphic on $U$. Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$. Then $u$ and $v$ are twice continuously differentiable, and $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$ on $U$. Thus $u$ and $v$ are harmonic functions on $U$.

Let $C$ be a circle with the centre $O$ and the radius $r > 0$ in the Euclidean plane $\mathbb{R}^2$. Let $T$ be a point of $C$, and let $\ell$ be a line through $T$. Then $\ell$ is tangent to $C$ at $T$ if and only if $\ell$ is perpendicular to the radius $OT$. That is, $(X - T) \cdot (T - O) = 0$ for each point $X$ of $\ell$.

Let $G$ be a group and let $g \in G$. If $g$ has finite order $n$, then $\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$, these $n$ elements are distinct, and thus $|\langle g \rangle| = n$. If $g$ has infinite order, then the powers $g^k$ for $k \in \mathbb{Z}$ are distinct, and $\langle g \rangle$ is infinite.

An event is a set of outcomes of a random experiment to which we give a probability. The event happens when the outcome of the experiment is in the set. The node event gives the precise definition.

A polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and corner points, the vertices. The word names the solid or its boundary surface. Cubes, pyramids and prisms are polyhedra.

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path. Let $a_1, \dots, a_n$ be distinct points of $U$, and let $f$ be holomorphic on $U \setminus \{a_1, \dots, a_n\}$. Let $\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \setminus \{a_1, \dots, a_n\}$. Then $$\oint_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^{n} n(\gamma, a_k) \operatorname{Res}(f, a_k),$$ where $n(\gamma, a_k)$ is the winding number of $\gamma$ around $a_k$ and $\operatorname{Res}(f, a_k)$ is the residue of $f$ at $a_k$.

Matrix multiplication is the operation that takes an $m \times n$ matrix and an $n \times p$ matrix and gives an $m \times p$ matrix. Entry $(i, k)$ of the product combines row $i$ of the first matrix with column $k$ of the second matrix.

A projection is a linear map from a vector space to itself that is equal to its own square. It projects the space onto its image along its kernel.

The nine-point circle of a triangle is the circle through the midpoints of its three sides. It also passes through the three feet of the altitudes and through the midpoints of the segments from the vertices to the orthocentre.

Let $n$ and $k$ be natural numbers with $k \le n$. Then $$\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$$

The limit inferior of a sequence is the limit of the infima of its tails. It describes the smallest value near which the sequence lies infinitely often. The node limit inferior gives the precise definition.

A formal power series is an infinite sum $\sum a_n X^n$ treated as an algebraic object, without any convergence condition. Formal power series form a ring that is the natural home of generating functions. The node formal power series gives the precise definition.

Let $x$ and $y$ be real numbers. Then the modulus of the complex number $x + y i$ is $$|x + y i| = \sqrt{x^2 + y^2}.$$

Take the graph of a differentiable function of two variables. Its tangent plane at a point is the plane through that point that best approximates the graph nearby. It is the graph of the linear approximation of the function at the point.

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. Then $A$ is similar to an upper triangular matrix over $F$ if and only if the characteristic polynomial of $A$ splits over $F$. Here, to split means to be a product of linear factors. In particular, over an algebraically closed field, such as $\C$, each square matrix is similar to an upper triangular matrix.

The hyperbolic cosine is the average of $e^x$ and $e^{-x}$. It is an even function with minimum value $1$ at $0$. The node hyperbolic cosine gives the precise definition.

Let $p$ be an odd prime and let $a$ be an integer with $p \nmid a$. Reduce each of the integers $a, 2a, \dots, \frac{p-1}{2} a$ to its least positive residue modulo $p$. Let $n$ be the number of these residues that are greater than $p/2$. Then $\left(\frac{a}{p}\right) = (-1)^n$, where the left side is the Legendre symbol.

The Dirichlet convolution is a product of two arithmetic functions, given by a sum over the divisors of $n$. The definition of the Dirichlet convolution gives the precise form.

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n + 1}$ converges, and $$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.$$

A system of linear differential equations is an equation $x' = A(t) x + b(t)$ for a vector-valued function $x$, with a matrix $A(t)$ of coefficients. Its theory extends that of a single linear equation. See the definition.

Let $X$ be a compact metric space and let $C(X)$ be the set of continuous functions $X \to \mathbb{R}$, with the uniform metric $d_\infty(f, g) = \sup_{x \in X} |f(x) - g(x)|$. Let $\mathcal{F} \subseteq C(X)$. Then the closure of $\mathcal{F}$ in $C(X)$ is compact if and only if $\mathcal{F}$ is equicontinuous and uniformly bounded. Here uniformly bounded means that there is a real $M$ with $|f(x)| \le M$ for all $f \in \mathcal{F}$ and $x \in X$.

The Bernoulli distribution with parameter $p$ is the distribution of a random variable with two values. The value $1$ has probability $p$ and the value $0$ has probability $1 - p$. It is the distribution of one Bernoulli trial. The node Bernoulli distribution gives the precise definition.

Let $p$ be a prime number and let $R$ be a commutative ring of characteristic $p$. Then $(x + y)^p = x^p + y^p$ for all $x, y \in R$.

A parametric equation of a line describes its points as the values of a function of one real parameter $t$. The line in $\mathbb{R}^3$ through the point $(x_0, y_0, z_0)$ with the nonzero direction vector $(a, b, c)$ has the parametric equations $$x = x_0 + a t, \quad y = y_0 + b t, \quad z = z_0 + c t, \qquad t \in \mathbb{R}.$$ In vector form, the line through the point $p$ with the direction vector $v \neq 0$ is the set of the points $p + t v$ for $t \in \mathbb{R}$. The same form works in $\mathbb{R}^n$ for each $n \geq 2$.