Product

The product of all elements in the collection.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaProduct (mathematics)↗MathWorldProduct↗WikidataQ901718↗Britannicatopic/product-mathematics↗FungrimProduct✓↗Fungrim entry33f13a✓↗Wolfram LanguageProduct!↗

Product(collection)the product of all elements in the collection.

Product(body, Tuple(index, lo, hi))a symbolic product over a range, closed-form whenever body is a power of the index (a factorial power) or a power WITH the index in the exponent (folds to a single power via a closed-form sum of the exponent); nested Tuple clauses reduce inner-first, so an inner limit depending on the outer index still closes. With hi = PositiveInfinity and body a ratio of polynomials in the index of equal degree and leading coefficient, closes via a Gamma-function product over the roots whenever those roots are exact (rational, Gaussian, or a quadratic surd) and the series converges; declines otherwise.

Product(body, Tuple(index))an index with neither bound stays unevaluated (an indefinite product is defined only up to a constant), where compute-engine read it as a degenerate range.

Typed signatures

Type	Package	Applies
(any, tuple*) -> number	analyticreplaces compute-engine	always
(any, tuple*) -> number	combinatoricsreplaces evaluation	always
(any, tuple*) -> number	compute-engine	always
(any, tuple*) -> number	evaluationreplaces compute-engine	always

Domain: Collections

Details

• The empty product is 1 by convention, matching $Factorial(0)$. See Factorial.
• The product of $1$ through $n$ is $n!$: $Product(Range(1, n)) = Factorial(n)$.
• Applies $\mathrm{Times}$ across the list's elements (the product of a list).
• Forces evaluation of a lazy collection like Range, which otherwise stays unevaluated on its own.
• A symbolic bound closes in two general shapes: $\prod i^m = (\prod i)^m$ for $m$ free of the index (a factorial power), and $\prod c^{f(i)} = c^{\sum f(i)}$ for $c$ free of the index, the sum found by Faulhaber's formula for any polynomial $f$. A product with more than one Tuple clause reduces inner-first, so a triangular product (an inner limit depending on the outer index) closes too, once the inner reduction leaves a shape the outer sum recognises.
• An infinite product $\prod_{k=k_0}^{\infty} P(k)/Q(k)$, for $P$ and $Q$ polynomials of equal degree and equal leading coefficient, factors both over $\mathbb{C}$ and rewrites the ratio as a Gamma-function product via the Pochhammer identity $\prod_{k=k_0}^{N}(k-a) = \Gamma(N+1-a)/\Gamma(k_0-a)$; the series converges exactly when the roots' sums agree, which cancels the $N \to \infty$ tail. Each factor then closes either straight to a factorial (a real integer root) or, paired with its negation, via Gamma recursion and the reflection formula $\Gamma(z)\Gamma(1-z) = \pi/\sin(\pi z)$. Roots come from Factor's own linear and irreducible-quadratic pieces (rational, Gaussian, or quadratic-surd only); anything past that -- an unresolved root, a pole, or an unpaired leftover -- leaves the product unevaluated rather than guess.

Examples

Basic #

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In$\prod\bigl\lbrack1, 2, 3, 4\bigr\rbrack$Out$24$

The empty product is 1 by convention, matching Factorial of 0

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In$\prod\bigl\lbrack\bigr\rbrack$Out$1$

$\prod_{i=1}^{6} i^2 = (6!)^2$

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In$\prod_{i=1}^6i^2$Out$518\,400$

A symbolic upper limit: $\prod_{i=1}^{n} i^2 = (n!)^2$

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In$\prod_{i=1}^{n}i^2$Out$(n!)^2$

$(6!)^2 = 720^2$

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In$\prod_{k=1}^6k^2$Out$518\,400$

A symbolic upper limit gives a closed form, $(n!)^2$

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In$\prod_{k=1}^{n}k^2$Out$(n!)^2$

An infinite product evaluated exactly: $\prod_{k\ge2}(1-k^{-4}) = \sinh\pi/(4\pi)$

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In$\prod_{k=2}^{∞}(1-\frac{1}{k^4})$Out$\frac{\sinh(π)}{4π}$

A double product with a dependent inner range collapses to one power of $2$

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In$\prod_{k=1, j=1}^{p, k}2^{j+k}$Out$2^{\frac{1}{2}p×(p+1)^2}$

Scope #

A symbolic upper limit: $\prod_{k=1}^{n} k = n!$. See Factorial

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In$\prod_{k=1}^{n}k$Out$n!$

Exponents add: $\prod_{k=1}^{n} x^k = x^{n(n+1)/2}$

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In$\prod_{k=1}^{n}x^{k}$Out$x^{\frac{n×(n+1)}{2}}$

A telescoping product: $\prod_{k=1}^{n} \frac{k+1}{k} = n+1$

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In$\prod_{k=1}^{n}\frac{k+1}{k}$Out$n+1$

An undefined function, expanded term by term

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In$\prod_{i=1}^4f(i)$Out$f(1)f(2)f(3)f(4)$

A step of 2 takes every other index

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In$\prod_{i=1}^4f(i)$Out$f(1)f(3)$

Over an explicit list of values

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In$\prod_{i∈\bigl\lbrack a, b, c\bigr\rbrack}f(i)$Out$f(a)f(b)f(c)$

A polynomial with roots 1, 2 and 3

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In$\prod_{k=1}^3(x-k)$Out$(x-3)(x-2)(x-1)$

A double product over a $3 \times 3$ grid

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In$\prod_{i=1, j=1}^{3, 3}(i+j)$Out$172\,800$

An inner limit that depends on the outer index, a triangular product

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In$\prod_{i=1, j=1}^{3, i}(i+j)$Out$2\,880$

A symbolic triangular product: $\prod_{i=1}^{p}\prod_{j=1}^{i} 2^{i+j} = 2^{p(p+1)^2/2}$

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In$\prod_{i=1, j=1}^{p, i}2^{i+j}$Out$2^{\frac{1}{2}p×(p+1)^2}$

An infinite product with a closed form: $\prod_{k=1}^{\infty} (1 + 1/k^2) = \frac{\sinh \pi}{\pi}$

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In$\prod_{k=1}^{∞}(1+\frac{1}{k^2})$Out$\frac{\sinh(π)}{π}$

An infinite telescoping product: $\prod_{k=2}^{\infty} (1 - 1/k^2) = \frac{1}{2}$

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In$\prod_{k=2}^{∞}(1-\frac{1}{k^2})$Out$\frac{1}{2}$

A quartic infinite product, via a Gamma product over a conjugate pair of Gaussian-integer roots $\pm i$ alongside the real roots $\pm 1$: $\prod_{i=2}^{\infty} (1 - 1/i^4) = \frac{\sinh \pi}{4\pi}$

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In$\prod_{i=2}^{∞}(1-\frac{1}{i^4})$Out$\frac{\sinh(π)}{4π}$

Exact rationals in a collection

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In$\prod\bigl\lbrack\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\bigr\rbrack$Out$\frac{1}{4}$

Symbolic elements

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In$\prod\bigl\lbrack a, b, c\bigr\rbrack$Out$abc$

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In$\prod_{k}^{n}a^{k}$Out$a^{\frac{n×(n+1)}{2}}$

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In$\prod_{j, k}^{m, j}a^{k}$Out$a^{\frac{1}{6}m×(m+1)(m+2)}$

Without bounds the nested product has no value and stays unevaluated

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In$\prod_{k, j}jk$Out$\prod_{k}\prod_{j}jk$

The unbounded inner product blocks evaluation; only nesting is made explicit

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In$\prod_{k=1, j}^{n}jk$Out$\prod_{k=1}^{n}\prod_{j}jk$

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In$\prod_{k}(3q^{2k}+5)$Out$\prod_{k}(3q^{2k}+5)$

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In$\prod_{k}\frac{q^{8(k+1)}+11}{q^{8k}+11}$Out$\prod_{k}\frac{q^{8(k+1)}+11}{q^{8k}+11}$

Applications #

Range stays lazy on its own, but Product forces it to compute $6!$

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In$\prod(1..6)$Out$720$

Properties #

The product of $1$ through $n$ is $n!$. See Factorial

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In$\prod\bigl\lbrack1, 2, 3, 4\bigr\rbrack=4!$Out$\mathrm{True}$

Neat examples #

The Wallis product, via the same Gamma-product method: $\prod_{k=1}^{\infty} \frac{4k^2}{4k^2-1} = \frac{\pi}{2}$

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In$\prod_{k=1}^{∞}\frac{4k^2}{4k^2-1}$Out$\frac{π}{2}$

Roots need not pair off as $\pm r$ -- $0$ and $-3$ each close on their own, straight to a factorial: $\prod_{n=1}^{\infty} \frac{n(n+3)}{(n+1)(n+2)} = \frac{1}{3}$

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In$\prod_{n=1}^{∞}\frac{n×(n+3)}{(n+1)(n+2)}$Out$\frac{1}{3}$

About

Examples	33
Checked against	Wolfram Language 28 agree4 differSymPy 15 agreemathlib4 2 agreeSageMath 5 agree1 differ2 unevaluated4 error
Crosswalk	WikipediaMathWorldWikidataBritannicaFungrimFungrim entryWolfram LanguageSageMathSymPy

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

As Sum; an empty range is 1, as ours.

See also: Length