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#### Search: #### Product

Function: product (<expr>, <i>, <i_0>, <i_1>) Represents a product of the values of <expr> as the index <i> varies from <i_0> to <i_1>. The noun form product is displayed as an uppercase letter pi. `product` evaluates <expr> and lower and upper limits <i_0> and <i_1>, `product` quotes (does not evaluate) the index <i>.

If the upper and lower limits differ by an integer, <expr> is evaluated for each value of the index <i>, and the result is an explicit product.

Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable `simpproduct` is `true`, additional rules are applied. In some cases, simplification yields a result which is not a product; otherwise, the result is a noun form product.

See also `nouns` and `evflag`.

Examples:

```          (%i1) product (x + i*(i+1)/2, i, 1, 4);
(%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
(%i2) product (i^2, i, 1, 7);
(%o2)                       25401600
(%i3) product (a[i], i, 1, 7);
(%o3)                 a  a  a  a  a  a  a
1  2  3  4  5  6  7
(%i4) product (a(i), i, 1, 7);
(%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
(%i5) product (a(i), i, 1, n);
n
/===\
! !
(%o5)                       ! !  a(i)
! !
i = 1
(%i6) product (k, k, 1, n);
n
/===\
! !
(%o6)                         ! !  k
! !
k = 1
(%i7) product (k, k, 1, n), simpproduct;
(%o7)                          n!
(%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
n
/===\
! !    1
(%o8)                       ! !  -----
! !  k + 1
k = 1
(%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
15  40
(%o9)                        a   b```

There are also some inexact matches for `product`. Try `?? product` to see them.

```(%o1)                                true
(%i2) ```

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