# Symbols:Greek/Sigma

## Contents

## Sigma

The $18$th letter of the Greek alphabet.

- Minuscules: $\sigma$ and $\varsigma$

- Majuscule: $\Sigma$

The $\LaTeX$ code for \(\sigma\) is `\sigma`

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The $\LaTeX$ code for \(\varsigma\) is `\varsigma`

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The $\LaTeX$ code for \(\Sigma\) is `\Sigma`

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### 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**.

### Summation

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 $S$.

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

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

The $\LaTeX$ code for \(\displaystyle \sum_{j \mathop = 1}^n a_j\) is `\displaystyle \sum_{j \mathop = 1}^n a_j`

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The $\LaTeX$ code for \(\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j\) is `\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j`

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

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### Sigma Function

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

Let $n$ be an integer such that $n \ge 1$.

The **sigma function** $\map \sigma n$ is defined on $n$ as being the sum of all the positive integer divisors of $n$.

That is:

- $\displaystyle \map \sigma n = \sum_{d \mathop \divides n} d$

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

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

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### 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.

### 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:

- $\sigma = \dfrac m A$

where: