Product

Product[f,{i,imax}]

evaluates the product .

Product[f,{i,imin,imax}]

starts with .

Product[f,{i,imin,imax,di}]

uses steps di.

Product[f,{i,{i1,i2,}}]

uses successive values i1, i2, .

Product[f,{i,imin,imax},{j,jmin,jmax},]

evaluates the multiple product .

Product[f,i]

gives the indefinite product .

Details and Options

  • Product[f,{i,imax}] can be entered as f.
  • can be entered as prod or \[Product].
  • Product[f,{i,imin,imax}] can be entered as f.
  • The limits should be underscripts and overscripts of in normal input, and subscripts and superscripts when embedded in other text.
  • Product uses the standard Wolfram Language iteration specification.
  • The iteration variable i is treated as local, effectively using Block.
  • If the range of a product is finite, i is typically assigned a sequence of values, with f being evaluated for each one.
  • In multiple products, the range of the outermost variable is given first.
  • The limits of a product need not be numbers. They can be Infinity or symbolic expressions.
  • If a product cannot be carried out explicitly by multiplying a finite number of terms, Product will attempt to find a symbolic result. In this case, f is first evaluated symbolically.
  • The indefinite product is defined so that the ratio of terms with successive gives .
  • Definite and indefinite summation can be mixed in any order.
  • For sums, the following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
    GeneratedParametersNonehow to name generated parameters
    MethodAutomaticmethod to use
    RegularizationNonewhat regularization to use
    VerifyConvergenceTruewhether to verify convergence
  • Possible values for Regularization include: None and "Dirichlet". {reg1,reg2,} specifies different schemes for different variables in a multiple product.
  • Product can do essentially all products that are given in standard books of tables.
  • Product is output in StandardForm using .

Examples

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Basic Examples  (5)

Numeric product:

Symbolic product:

Use prod to enter and to enter the lower limit, then for the upper limit:

Infinite product:

Multiple product with product over performed first:

Scope  (28)

Basic Uses  (9)

A definite product over a finite range:

Use step size 2:

Use a list of elements:

Plot the sequence of partial products:

A multiple product over finite ranges:

Use different step sizes:

Plot a multivariate sequence and the logarithm of its partial products:

The outermost product bounds can depend on inner variables:

Combine a product over lists with standard iteration ranges:

The elements in the iterator list can be any expression:

Compute a product over an infinite range:

Multivariate product over infinite ranges:

Use a symbolic range:

Indefinite products:

The ratio is equivalent to the multiplicand:

The definite product is given as the ratio of indefinite products:

Multivariate indefinite products:

Mixes of indefinite and definite products:

Use GenerateConditions to get the conditions under which the answer is true:

Refine the resulting answer:

Use Assumptions to provide assumptions directly to Product:

Applying N to an unevaluated product effectively uses NProduct:

Special Indefinite Products  (10)

Ratios of expressions with a general function:

Indefinite products are unique up to a constant factor:

For exponential functions, products are equivalent to sums :

The results differ by a constant factor:

The product of polynomial functions can always be done in terms of factorial functions:

Products of rational functions can always be represented as rational functions and factorials:

A minimal number of factorial functions will be used:

Hypergeometric term sequences can be represented in terms of BarnesG:

The DiscreteRatio is rational for all hypergeometric term sequences:

Many functions give hypergeometric terms:

Any products of hypergeometric terms are hypergeometric terms:

Their products in general require BarnesG:

Q-polynomial products can always be represented in terms of q-factorial functions:

A q-polynomial is the composition of a polynomial with an exponential:

Products of q-rational functions can always be done in terms of q-rational and q-factorials:

A q-rational function is the composition of a rational function with an exponential:

In general, Root objects are needed:

Polynomials and rational functions of trigonometric functions:

Similarly for hyperbolic functions:

Rational functions raised to a polynomial power:

Floor and Ceiling related functions:

Periodic sequences:

Any function applied to a periodic sequence generates a periodic sequence:

A sequence raised to a periodic exponent:

A periodic sequence raised to a nonperiodic exponent:

Telescoping products:

Special Definite Products  (9)

For exponential functions products are equivalent to sums :

Rational products can be represented as factorial functions:

For infinite products, the limit of the multiplicand needs to be 1:

An infinite product may not converge:

Hypergeometric term products can be represented in terms of BarnesG:

Q-polynomial products can be represented in terms of q-factorial functions:

Some products of q-rational functions can be represented as q-rational functions:

But in general they require q-factorial functions:

Products of trigonometric and hyperbolic functions:

Piecewise products can often be reduced to the previous classes:

In other cases, the piecewise part is eventually constant:

Special products:

Telescoping products:

Multiple products:

Options  (4)

Assumptions  (1)

Use Assumptions to study the behavior under different conditions on the parameter:

GenerateConditions  (1)

Generate conditions for convergence of an infinite product:

Regularization  (1)

The product of all primes is divergent:

Use Regularization to assign a finite value to this product:

VerifyConvergence  (1)

The product of all natural numbers is divergent:

Setting VerifyConvergence to False may assign a finite value in such cases:

Applications  (6)

Functions Defined by a Product  (1)

Many special functions are defined by products, including Factorial:

Pochhammer:

FactorialPower:

QPochhammer:

Hyperfactorial:

And BarnesG:

Product Representations for Constants  (1)

Many constants can be represented as products including Pi:

E:

Glaisher:

Product Representations for Functions  (1)

Weierstrass factorizations for elementary functions:

Approximate Representation Using Lagrange Interpolation  (1)

A Lagrange interpolating polynomial:

A Lagrange interpolating polynomial for symbolic values:

A Newton interpolating polynomial for symbolic values:

Optimized Representations from Product  (1)

Using a symbolic closedform product can provide fast evaluation:

Compare timings of the closed form to procedural evaluation:

Likelihood Functions for Parameter Estimation  (1)

Define a likelihood function for a distribution and dataset:

The likelihood function for an ExponentialDistribution for a small dataset:

Find the maximum-likelihood parameter estimator:

Likelihood function for BernoulliDistribution:

Properties & Relations  (4)

NProduct will use numerical methods to compute products:

Applying N to an unevaluated product effectively uses NProduct:

DiscreteRatio is the inverse for indefinite products:

Product essentially solves a special difference equation as solved by RSolve:

Possible Issues  (2)

A product may not be convergent:

The upper product limit is assumed to be an integer distance from the lower limit:

Use GenerateConditions to get explicit assumptions:

In general, the upper limit is assumed to be a multiple of distance from the lower limit:

With GenerateConditions, the assumptions are explicit:

Neat Examples  (2)

Answers in terms of Zeta and PolyLog derivatives:

Create a gallery of infinite products:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)
2008
 (7.0)
2019
 (12.0)