Fungrim entry: 1e2755

Symbol: Product —

\prod_{n} f(n)

— Product

Product(S), rendered as

\prod S

, gives the product of the elements of the set

S

. The product is required to be absolutely convergent. The empty product is 1.

Product(f(n), For(n, a, b)), rendered as

\prod_{n=a}^{b} f(n)

, gives the product of

f(n)

for integers

n

from

a

b

, where

a

and

b

should be integers or

-\infty

\infty

. If

a < b

, the product is empty. The product

\prod_{n=0}^{\infty} f(n)

is interpreted as

\lim_{N \to \infty} \prod_{n=0}^{N} f(n)

and can be conditionally convergent.

Product(f(n), For(n, a, b), P(n)), rendered as

\prod_{\textstyle{n=a \atop P(n)}}^{b} f(n)

, is as above but only terms satisfying the predicate

P(n)

are included.

Product(f(x), ForElement(x, S)), rendered as

\prod_{x \in S} f(x)

, gives the product of

f(x)

for all

x

in the set

S

. The product is required to be absolutely convergent.

Product(f(x), ForElement(x, S), P(x)), rendered as

\prod_{\textstyle{x \in S \atop P(x)}} f(x)

, gives the product of

f(x)

for all

x

in the set

S

and satisfying the predicate

P(x)

. The product is required to be absolutely convergent.

Product(f(x), For(x), P(x)), rendered as

\prod_{P(x)} f(x)

, gives the product of

f(x)

for all

x

satisfying the predicate

P(x)

. The predicate

P(x)

should define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. The product is required to be absolutely convergent.

The empty product is 1. The product can range over an uncountable number of factors, as long as only countably many factors are nonzero.

The special expression For(x), For(x, a, b) or ForElement(x) declares x as a locally bound variable within the scope of the arguments to this operator. Similarly For(Tuple(x, y)), ForElement(Tuple(x, y), S) etc. defines multiple locally bound variables.

Definitions:

Fungrim symbol	Notation	Short description
`Product`	$\prod_{n} f(n)$	Product
`Infinity`	$\infty$	Positive infinity
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence

Source code for this entry:

Entry(ID("1e2755"),
    SymbolDefinition(Product, Product(f(n), For(n)), "Product"),
    Description(SourceForm(Product(S)), ", rendered as ", Product(S), ", gives the product of the elements of the set", S, ". The product is required to be absolutely convergent. The empty product is 1."),
    Description(SourceForm(Product(f(n), For(n, a, b))), ", rendered as ", Product(f(n), For(n, a, b)), ", gives the product of", f(n), "for integers", n, "from", a, "to", b, ", where", a, "and", b, "should be integers or", Neg(Infinity), "or", Infinity, ". ", "If", Less(a, b), ", the product is empty. The product", Product(f(n), For(n, 0, Infinity)), "is interpreted as", SequenceLimit(Product(f(n), For(n, 0, N)), Var(N), Infinity), "and can be conditionally convergent."),
    Description(SourceForm(Product(f(n), For(n, a, b), P(n))), ", rendered as ", Product(f(n), For(n, a, b), P(n)), ", is as above but", "only terms satisfying the predicate", P(n), "are included."),
    Description(SourceForm(Product(f(x), ForElement(x, S))), ", rendered as ", Product(f(x), ForElement(x, S)), ", gives the product of", f(x), "for all", x, "in the set", S, ". The product is required to be absolutely convergent."),
    Description(SourceForm(Product(f(x), ForElement(x, S), P(x))), ", rendered as ", Product(f(x), ForElement(x, S), P(x)), ", gives the product of", f(x), "for all", x, "in the set", S, "and satisfying the predicate", P(x), ". The product is required to be absolutely convergent."),
    Description(SourceForm(Product(f(x), For(x), P(x))), ", rendered as ", Product(f(x), For(x), P(x)), ", gives the product of", f(x), "for all", x, "satisfying the predicate", P(x), ". ", "The predicate", P(x), "should define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "The product is required to be absolutely convergent."),
    Description("The empty product is", 1, ". The product can range over an uncountable number of factors, as long as only countably many factors are nonzero."),
    Description("The special expression", SourceForm(For(x)), ", ", SourceForm(For(x, a, b)), " or", SourceForm(ForElement(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "Similarly", SourceForm(For(Tuple(x, y))), ", ", SourceForm(ForElement(Tuple(x, y), S)), "etc. defines multiple locally bound variables."))

Topics using this entry

Operators