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Product

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Product[f, {i, imax}]
    gives the product of f for i from 1 to imax.

Product[f, {i, imin, imax}], Product[f, {i, imin, imax, di}] and Product[f, {i, {i1, i2, ...}}] use the standard iterator forms; multiple iterators give nested products (an inner bound may depend on an outer index). Product[f, i] gives the indefinite product (anti-quotient). The index is localised (HoldAll). Finite ranges are multiplied out directly; symbolic, indefinite and convergent infinite products are evaluated in exact closed form (n!, Pochhammer, Gamma ratios, base^k, QPochhammer, BarnesG) via a Method polyalgorithm.

Options: Method (Automatic | "Telescoping" | "Rational" | "Geometric" | "QProduct"), VerifyConvergence (default True; a divergent infinite product gives Product::div), GenerateConditions, Assumptions. N[Product[...]] routes to NProduct.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Product[k, {k, 1, n}]
Out[1]= Factorial[n]

In[2]:= Product[k + a, {k, 1, n}]
Out[2]= Pochhammer[1 + a, n]

In[3]:= Product[2^k, {k, 1, n}]
Out[3]= 2^(1/2 n (1 + n))

In[4]:= Product[1 + 1/k^2, {k, 1, Infinity}]
Out[4]= Sinh[Pi]/Pi

In[5]:= Product[1 - a q^k, {k, 0, n - 1}]
Out[5]= QPochhammer[a, q, n]

In[6]:= Product[k^k, {k, 1, n}]
Out[6]= Hyperfactorial[n]

In[1]:= Product[(k^2 - 1)/(k^2 + 1), {k, 2, Infinity}]
Out[1]= Pi Csch[Pi]

In[2]:= Product[(k^3 - 1)/(k^3 + 1), {k, 2, Infinity}]
Out[2]= 2/3

Implementation notes

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References