# Symbols:Greek/Pi

## Pi

The $16$th letter of the Greek alphabet.

- Minuscules: $\pi$ and $\varpi$

- Majuscule: $\Pi$

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

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

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

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### Constant

- $\pi$

The real number $\pi$ (pronounced **pie**) is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$

### Prime-Counting Function

- $\map \pi x$

The **prime-counting function** is the function $\pi: \R \to \Z$ which counts the number of primes less than or equal to some real number.

That is:

- $\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$

where $\mathbb P$ denotes the set of prime numbers.

The $\LaTeX$ code for \(\map \pi x\) is `\map \pi x`

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### Projection

- $\pi_i$

The notation $\pi_i$ is often used for the $i$th projection.

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

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### Rectangle Function

- $\map \Pi x$

The **rectangle function** is the real function $\Pi: \R \to \R$ defined as:

- $\forall x \in \R: \map \Pi x := \begin {cases} 1 : & \size x \le \dfrac 1 2 \\ 0 : & \size x > \dfrac 1 2 \end {cases}$

The $\LaTeX$ code for \(\map \Pi x\) is `\map \Pi x`

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### Probability Generating Function

- $\map {\Pi_X} s$

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

The **probability generating function for $X$**, denoted $\map {\Pi_X} s$, is the formal power series defined by:

- $\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$

The $\LaTeX$ code for \(\map {\Pi_X} s\) is `\map {\Pi_X} s`

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### Product Notation

- $\ds \prod_{j \mathop = 1}^n a_j$

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

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

The composite is called the **product** of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

- $\ds \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$

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

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

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The $\LaTeX$ code for \(\ds \prod_{\map \Phi j} a_j\) is `\ds \prod_{\map \Phi j} a_j`

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## Linguistic Note

While the conventional contemporary prounciation of $\pi$ in Western English is **pie**, it is worth noting that the "correct" Greek pronunciation of the name of the letter $\pi$ is in fact the same as the letter **p** is pronounced in English.

It is just as well that $\pi$ is pronounced **pie**, otherwise the opportunity for confusion between $\pi$ and $p$ in spoken language would be too great.