Symbols:Greek

Gamma

Gamma Function

$\Gamma \left({z}\right)$

The Gamma function $\Gamma: \C \to \C \ $ is defined, for the open right half-plane, as:

$\displaystyle \Gamma \left({z}\right) = \int_0^\infty t^{z-1} e^{-t} \mathrm d t$

and for all other values of $z$ except the non-positive integers as:

$\Gamma \left({z + 1}\right) = z \Gamma \left({z}\right)$

Other equivalent definitions exist, as follows.

Weierstrass Form

Of note is the Weierstrass form:

$\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n=1}^\infty \left({\left({ 1 + \frac z n}\right) e^{\frac {-z} n}}\right)$

where $\gamma$ is the Euler-Mascheroni constant. The Weierstrass expression is valid for all $\C$.

Euler Form

Another important form of the Gamma function is the Euler form:

$\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n=1}^\infty \left({ \left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \to \infty} \frac {m^z m!} {z \left({z+1}\right) \left({z+2}\right) \ldots \left({z+m}\right)}$

which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\} \ $.

The $\LaTeX$ code for $\Gamma \left({z}\right)$ is \Gamma \left({z}\right).

The Euler-Mascheroni Constant

$\gamma$

The Euler-Mascheroni Constant $\gamma$ is the real number that is defined as:

$\displaystyle \gamma := \lim_{n \to \infty} \left({\sum_{k=1}^n \frac 1 k - \ln n}\right)$

The existence of this constant is demonstrated in Existence of Euler-Mascheroni Constant.

Its value is approximately $0.57721\ 56649\ 01532\ 86060\ 6512 \ldots$ This sequence is A001620 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The $\LaTeX$ code for $\gamma$ is \gamma.

Delta

Diagonal Relation

$\Delta_S$

Let $S$ be a set.

The diagonal relation on $S$ is a relation on $S$ such that:

$\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\} \subseteq S \times S$

The $\LaTeX$ code for $\Delta_S$ is \Delta_S.

Product of Differences

$\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$

Let $n \in \Z, n > 0$ be an integer.

Then $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ is defined as:

$\displaystyle \Delta_n = \prod_{1 \le i < j \le n} \left({x_i - x_j}\right)$

Thus $\Delta_n$ is the product of the difference of all pairs of $\left\{{x_1, x_2, \ldots, x_n}\right\}$ where the index of the first is less than the index of the second.

The $\LaTeX$ code for $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ is \Delta_n \left({x_1, x_2, \ldots, x_n}\right).

Kronecker Delta

$\delta_{x y}$

Let $\Gamma$ be a set.

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is defined as:

$\forall \left({\alpha, \beta}\right) \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & :\alpha = \beta \\ 0_R & :\alpha \ne \beta \end{cases}$

This use of $\delta$ is known as the Kronecker delta notation or Kronecker delta convention.

It can be expressed in Iverson bracket notation as:

$\delta_{\alpha \beta} := \left[{\alpha = \beta}\right]$

The $\LaTeX$ code for $\delta_{x y}$ is \delta_{x y}.

Change

$\Delta x_n$

$\Delta$ is often used to mean change or difference.

For example, for the definition of slope:

$\dfrac {\Delta y}{\Delta x} = \dfrac {y_2-y_1}{x_2-x_1} = \dfrac {\text{change in } y}{\text{change in } x}$

The $\LaTeX$ code for $\Delta x_n$ is \Delta x_n.

Epsilon

Element of a Set

$x \in S$, $S \owns x$

See element of a set.

The $\LaTeX$ code for $\in$ is \in.

The $\LaTeX$ code for $\owns$ is \ni (that is, in backwards), or \owns.

A small positive quantity

Many a proof in analysis will famously start:

"Let $\epsilon > 0$ ..."

where it is frequently left unstated that $\epsilon$ is a real number, arbitrarily small.

The $\LaTeX$ code for $\epsilon > 0$ is \epsilon > 0.

Alternative Symbol

While $\epsilon$ is common, so is $\varepsilon$. The symbols are, in general, interchangeable.

Some writers prefer $\epsilon$ and some prefer $\varepsilon$.

The $\LaTeX$ code for $\varepsilon$ is \varepsilon.

Iota

Inclusion Mapping

Used by some sources to denote the mapping on $S$ to $T$ where $S \subseteq T$:

$\iota_S: S \to T: \forall x \in S: \iota_S \left({x}\right) = x$

The $\LaTeX$ code for $\iota_S$ is \iota_S.

Identity Arithmetic Function

The identity arithmetic function $\iota: S \to \Z$ is defined for $n \geq 1$ by:

$\forall n \in S: \iota \left({n}\right) = \delta_{n1}$

where:

$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$.

$\delta$ is the Kronecker delta.

That is:

$\forall n \in S: \iota \left({n}\right) = \begin{cases} 1 & : n = 1\\ 0 & : n \ne 1 \end{cases}$

The $\LaTeX$ code for $\iota \left({n}\right)$ is \iota \left({n}\right).

Lambda

Von Mangoldt Function

$\Lambda \left({n}\right)$

The Von Mangoldt function (also known as the Mangoldt function) $\Lambda: \N \to \R$ is defined as:

$\Lambda \left({n}\right) = \begin{cases} \ln p & : \exists m \in \N, p \in \mathbb P: n = p^m \\ 0 & : \text{otherwise} \end{cases}$

where $\mathbb P$ is the set of all prime numbers.

The $\LaTeX$ code for $\Lambda \left({n}\right)$ is \Lambda \left({n}\right).

Linear Density

$\lambda$

Used to denote the linear density of a given one-dimensional body:

$\displaystyle \lambda = \frac m l$

where:

• $m$ is the body's mass;
• $l$ is the body's length.

The $\LaTeX$ code for $\lambda$ is \lambda.

Parameter of Poisson Distribution

$\lambda$

Used to denote the parameter of a given Poisson distribution:

Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Then $X$ has the poisson distribution with parameter $\lambda$ (where $\lambda > 0$) if:

• $\operatorname{Im} \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N$

• $\displaystyle \Pr \left({X = k}\right) = \frac 1 {k!} \lambda^k e^{-\lambda}$

The $\LaTeX$ code for $\lambda$ is \lambda.

Mu

Expectation

$\mu$

Often used to denote the expectation of a given random variable.

The $\LaTeX$ code for $\mu$ is \mu.

Linear Density

$\mu$

Used to denote the linear density of a given one-dimensional body:

$\displaystyle \mu = \frac m l$

where:

• $m$ is the body's mass;
• $l$ is the body's length.

The $\LaTeX$ code for $\mu$ is \mu.

Parameter of Poisson Distribution

$\mu$

Used as an alternative to $\lambda$ to denote the parameter of a given Poisson distribution.

The $\LaTeX$ code for $\mu$ is \mu.

Moment of Discrete Random Variable

$\mu'_n$

Let $X$ be a discrete random variable.

Then the $n$th moment of $X$ is denoted $\mu'_n$ and defined as:

$\mu'_n = E \left({X^n}\right)$.

where $E$ denotes the expectation function.

The $\LaTeX$ code for $\mu'_n$ is \mu'_n.

Pi

Real Constant

$\pi$

The real number $\pi$ (pronounced pie) is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$ This sequence is A000796 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The $\LaTeX$ code for $\pi$ is \pi.

Probability Generating Function

$\Pi_X \left({s}\right)$

Let $X$ be a discrete random variable whose codomain is a subset of $\N = \left\{{0, 1, 2, \ldots}\right\}$.

The probability generating function (p.g.f.) for (or of) $X$ is denoted $\Pi_X \left({s}\right)$ and defined as:

$\Pi_X \left({s}\right) = E \left({s^X}\right)$

where:

• $s$ is a dummy variable;
• $E \left({s^X}\right)$ is the expectation of $s^x$ for $x \in X$.

The $\LaTeX$ code for $\Pi_X \left({s}\right)$ is \Pi_X \left({s}\right).

Product Notation

Let $\left({S, \times}\right)$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Then the composite is called the product of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:

$\displaystyle \prod_{j=1}^n a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

Alternatively:

$\displaystyle \prod_{1 \le j \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:

$\displaystyle \prod_{\Phi \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

The $\LaTeX$ code for $\displaystyle \prod_{j=1}^n a_j$ is \displaystyle \prod_{j=1}^n a_j.

The $\LaTeX$ code for $\displaystyle \prod_{1 \le j \le n} a_j$ is \displaystyle \prod_{1 \le j \le n} a_j.

The $\LaTeX$ code for $\displaystyle \prod_{\Phi \left({j}\right)} a_j$ is \displaystyle \prod_{\Phi \left({j}\right)} a_j.

Rho

Density

$\rho$

Used to denote the density of a given body:

$\displaystyle \rho = \frac m V$

where:

• $m$ is the body's mass;
• $V$ is the body's volume.

The $\LaTeX$ code for $\rho$ is \rho.

Area Density

$\rho_A$

Used to denote the area density of a given two-dimensional body:

$\displaystyle \rho_A = \frac m A$

where:

• $m$ is the body's mass;
• $A$ is the body's area.

The $\LaTeX$ code for $\rho_A$ is \rho_A.

Sigma

Event Space

$\Sigma$

Let $\mathcal E$ be an experiment.

The event space of $\mathcal E$ is usually denoted $\Sigma$ (Greek capital sigma), and is the set of all outcomes of $\mathcal E$ which are interesting.

The $\LaTeX$ code for $\Sigma$ is \Sigma.

Sum Notation

Let $\left({S, +}\right)$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Then the composite is called the sum of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:

$\displaystyle \sum_{j=1}^n a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$

Alternatively:

$\displaystyle \sum_{1 \le j \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$

If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:

$\displaystyle \sum_{\Phi \left({j}\right)} a_j = \text{ The sum of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

The $\LaTeX$ code for $\displaystyle \sum_{j=1}^n a_j$ is \displaystyle \sum_{j=1}^n a_j.

The $\LaTeX$ code for $\displaystyle \sum_{1 \le j \le n} a_j$ is \displaystyle \sum_{1 \le j \le n} a_j.

The $\LaTeX$ code for $\displaystyle \sum_{\Phi \left({j}\right)} a_j$ is \displaystyle \sum_{\Phi \left({j}\right)} a_j.

Sigma Function

$\sigma \left({n}\right)$

Let $n$ be an integer such that $n \ge 2$.

The sigma function $\sigma \left({n}\right)$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.

That is:

$\displaystyle \sigma \left({n}\right) = \sum_{d \backslash n} d$

where $\displaystyle \sum_{d \backslash n}$ is the sum over all divisors of $n$.

The $\LaTeX$ code for $\sigma \left({n}\right)$ is \sigma \left({n}\right).

Surface Charge Density

$\sigma$

Used to denote the surface charge density of a given body:

$\displaystyle \sigma = \frac q A$

where:

• $q$ is the body's electric charge;
• $A$ is the body's area.

The $\LaTeX$ code for $\sigma$ is \sigma.

Area Density

$\sigma$

Used sometimes, although $\rho_A$ (Greek letter rho) is more common, to denote the area density of a given two-dimensional body:

$\displaystyle \sigma = \frac m A$

where:

• $m$ is the body's mass;
• $A$ is the body's area.

The $\LaTeX$ code for $\sigma$ is \sigma.

Tau

Tau Function

$\tau \left({n}\right)$

Let $n$ be an integer such that $n \ge 2$.

The tau function $\tau \left({n}\right)$ is defined on $n$ as being the total number of positive integer divisors of $n$.

That is:

$\displaystyle \tau \left({n}\right) = \sum_{d \backslash n} 1$

where $\displaystyle \sum_{d \backslash n}$ is the sum over all divisors of $n$.

The $\LaTeX$ code for $\tau \left({n}\right)$ is \tau \left({n}\right).

Phi

Euler Phi Function

$\phi \left({n}\right)$

Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The totient, indicator or Euler $\phi$-function is the function $\phi: \Z_{>0} \to \Z_{>0}$ defined as:

$\phi \left({n}\right) = $ the number of integers less than or equal to $n$ which are prime to $n$

That is:

$\phi \left({n}\right) = \left|{S_n}\right|: S_n = \left\{{k: 1 \le k \le n, k \perp n}\right\}$

The $\LaTeX$ code for $\phi \left({n}\right)$ is \phi \left({n}\right).

Chi

Characteristic Function

$\chi_E$

Let $E \subseteq S$.

The characteristic function of $E \ $ is the function $\chi_E: S \to \left\{{0, 1}\right\}$ defined as:

$\chi_E \left({x}\right) = \begin{cases} 1 & : x \in E \\ 0 & : x \notin E \end{cases}$

Alternatively, and equivalently, it can be written as:

$\chi_E \left({x}\right) = \begin{cases} 1 & : x \in E \\ 0 & : x \in \complement_S \left({E}\right) \end{cases}$

The $\LaTeX$ code for $\chi_E$ is \chi_E.

Omega

Sample Space

$\Omega$

Let $\mathcal E$ be an experiment.

The sample space of $\mathcal E$ is usually denoted $\Omega$ (Greek capital omega), and is defined as the set of all possible outcomes of $\mathcal E$.

The $\LaTeX$ code for $\Omega$ is \Omega.

Elementary Event

$\omega$

Let $\mathcal E$ be an experiment.

An elementary event of $\mathcal E$, often denoted $\omega$ (Greek lowercase omega) is one of the elements of the sample space $\Omega$ (Greek capital omega) of $\mathcal E$.

The $\LaTeX$ code for $\omega$ is \omega.

Order Type of Natural Numbers

$\omega$

The order type of $\left({\N, \le}\right)$ is denoted $\omega$ (omega).

The $\LaTeX$ code for $\omega$ is \omega.

Greek Alphabet

Position	Lowercase	Uppercase	Name	Some mathematical uses
1	$\alpha$	$\textrm{A}$	Alpha
2	$\beta$	$\textrm{B}$	Beta
3	$\gamma$	$\Gamma$	Gamma
4	$\delta$	$\Delta$	Delta	Lowercase: $\varepsilon -\delta$ proofs. Uppercase: differences.
5	$\epsilon$	$\textrm{E}$	Epsilon
6	$\zeta$	$\textrm{Z}$	Zeta
7	$\eta$	$\textrm{H}$	Eta
8	$\theta$	$\Theta$	Theta	commonly used to denote an angle or morphism
9	$\iota$	$\textrm{I}$	Iota
10	$\kappa$	$\textrm{K}$	Kappa
11	$\lambda$	$\Lambda$	Lambda	Lowercase: eigenvalues
12	$\mu$	$\textrm{M}$	Mu
13	$\nu$	$\textrm{N}$	Nu
14	$\xi$	$\Xi$	Xi
15	$o$	$\textrm{O}$	Omicron
16	$\pi$	$\Pi$	Pi
17	$\rho$	$\textrm{P}$	Rho
18	$\sigma$	$\Sigma$	Sigma
19	$\tau$	$\textrm{T}$	Tau
20	$\upsilon$	$\Upsilon$	Upsilon
21	$\phi$	$\Phi$	Phi
22	$\chi$	$\textrm{X}$	Chi
23	$\psi$	$\Psi$	Psi
24	$\omega$	$\Omega$	Omega
Lowercase variants
25	$\varepsilon$		Varepsilon	common epsilon for $\varepsilon -\delta$ proofs
26	$\vartheta$		Vartheta
27	$\varpi$		Varpi
28	$\varrho$		Varrho
29	$\varsigma$		Varsigma
30	$\varphi$		Varphi