product
Definite and indefinite products
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product(f
,i
) product(f
,i = a .. b
) product(f
,i = RootOf(p, <x>)
) product(f
,i in RootOf(p, <x>)
) product(f
,i in {x_{1}, x_{2}, …}
)
product(f, i)
computes the indefinite product
of f(i) with
respect to i,
i.e., a closed form g such
that .
product(f, i = a..b)
tries to find a closed
form representation of the product .
product (f, i = RootOf(p, x))
computes the
product of f(i) over
the roots of the polynomial p.
product(f, i in { x_{1}, x_{2},
…})
computes the product .
product
serves for simplifying symbolic products.
It should not be used for multiplying a finite
number of terms: if a
and b
are
integers of type DOM_INT
, the call _mult(f $ i
= a..b)
is more efficient than product(f, i = a..b)
.
product(f, i)
computes the indefinite product
of f
with respect to i
. This
is an expression g
such that .
It is implicitly assumed that i
runs through
integers only.
product(f, i = a..b)
computes the definite
product with i
running from a
to b
.
It is implicitly assumed that a ≤ b;
it is an error if this is inconsistent.
a
and b
must not be numbers
other than integers.
If ba
is an integer, the explicit product f(a) f(a +
1) … f(b) is
returned if it has no more than 1000 factors.
product (f, i = RootOf(p, x))
computes the
definite product with i
running through the roots
of the polynomial p
in x
according
to their multiplicity, i.e., the number of factors is equal to the
degree of p
.
The calls product (f, i = RootOf(p, x))
and product
(f, i in RootOf(p, x))
are equivalent.
The system returns a symbolic product
call
if it cannot compute a closed form representation of the product.
Each of the following two calls computes the product 1 2 3 4 5:
product(i, i = 1..5) = _mult(i $ i = 1..5)
However, using _mult
is usually more efficient when
the boundaries are integers of type DOM_INT
.
There is a closed form of this definite product from 1 to n:
product(i, i = 1..n)
Since the upper boundary is a symbolic identifier, _mult
cannot
handle this product:
_mult(i $ i = 1..n)
The corresponding indefinite product is:
product(i, i)
The indefinite and the definite product of 2 i + 1 are:
product(2*i + 1, i)
product(2*i + 1, i = 1..n)
The boundaries may be symbolic expressions or as well:
product(i^2/(i^2  1), i = 2..infinity)
The system cannot find closed forms of the following two products
and returns symbolic product
calls:
delete f: product(f(i), i)
product((1 + 2^(i)), i = 1..infinity)
An approximation can be computed numerically via float
:
float(%)
Alternatively, you can call numeric::product
directly. This is usually
more efficient, since it skips the symbolic computations performed
by product
:
numeric::product((1 + 2^(i)), i = 1..infinity)
Some products over the roots of a polynomial:
product(1 + 1/x, x = RootOf(x^2  5*x + 6))
product(r+c, r = RootOf(x^3 + a*x^2 + b*x + c, x))
The multiplicity of roots is taken into account:
product(x+2, x in RootOf(x^5))
MuPAD^{®} finds closed forms for products of rational expressions.
In other cases, a symbolic call to product
is returned:
product(sin(r), r = RootOf(x^2  PI^2/4, x))
An approximation can be computed numerically via float
:
float(%)
Some products over elements of a set:
product(x+2, x in {2,4,8})
product(a*x, x in {3, b, 5})
Identical objects appear only once in a set. Therefore, the
second a
in the following example has no effect
on the result:
product(x, x in {a,a,7,b})

An arithmetical
expression depending on 

The product index: an identifier or indexed indentifier 

The boundaries: arithmetical expressions 

A polynomial
expression in 

Indeterminate 
The product over the roots of a polynomial is computed via polylib::resultant
.