∏ (nary product)
Computes the product of all elements in a container or in a sequence. The elements may be numbers or square matrices.
Syntax

∏(X)

X
is a vector, matrix, list, or set


∏(expr, var, a, b)

expr
is an expression in one variablevar

var
is the variable inexpr

a
is the lower bound 
b
is the upper bound

Description
Computing the product of all elements in a container
If X
is a container, then ∏(X)
is the product of all elements in X
.
The following container types are supported:

vectors

the result is the product of all components


matrices

the result is the product of all entries


lists containing numbers

the result is the product of all numbers


lists containing square matrices

the result is the matrix product of all matrices


sets containing numbers

the result is the product of all numbers

Typewise, the factors may be either real or complex, and the result is complex iff at least one of the factors is complex.
Computing the product of all elements in a sequence
∏(expr, var, a, b)
computes the product of the expression expr
in one variable var
as var
takes all integer values from a
to b
(inclusively). This precisely implements the pi notation for nary products.
expr
must return a number or a matrix.
Notes
These note apply to both modes of operation: computing the product of all elements in a container and computing the product of all elements in a sequence.
The empty product is 1
. Notice that the product of an empty collection of matrices is also equal to the scalar 1
, because the system cannot tell that the collection indeed is an empty collection of matrices. And even if it could, it wouldn’t know the size of the matrices.
When computing the product of a collection of objects, the objects must be of the same kind (numbers or matrices). If they are matrices, they must all be of the same size.
Notice that ∏
is a function, not a prefix operator. Hence, to compute the product of all elements in a container X
you must write ∏(X)
– ∏X
would be a syntax error.
You cannot compute the product of all matrices in a set because matrix multiplication is not commutative and a set is not an ordered container.
Examples
Computing the product of all elements in a container
∏(❨2, 1, 3, −4, 5, 2, 1, 3, −2❩)
1440
∏(❨❨4, 2, 1❩, ❨3, 2, 5❩❩)
240
∏('(❨❨1, 3❩, ❨2, 1❩❩, ❨❨−2, 1❩, ❨0, 3❩❩, ❨❨1, 0❩, ❨−1, i❩❩))
⎛−12 10⋅i⎞ ⎝ −9 5⋅i ⎠
∏(unique(PrimeFactors(19871226)))
6623742
Computing the product of all elements in a sequence
abs(∏(1 − 1/prime(n)^2, n, 1, 100000) − 6/π^2)
3.11339510472⋅10^−8