From ProofWiki
Jump to: navigation, search

Previous  ... Next


The $16$th letter of the Greek alphabet.

Minuscules: $\pi$ and $\varpi$
Majuscule: $\Pi$

The $\LaTeX$ code for \(\pi\) is \pi .
The $\LaTeX$ code for \(\varpi\) is \varpi .

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

Real Constant

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

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:

$\displaystyle \forall x \in \R: \pi \left({x}\right) = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$

The $\LaTeX$ code for \(\pi \left({x}\right)\) is \pi \left({x}\right) .


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

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

Probability Generating Function

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

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 $\Pi_X \left({s}\right)$, is the formal power series defined by:

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

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 $S$.

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

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

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

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