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:

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:

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:

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

Generate conditions for convergence of an infinite product:

The product of all primes is divergent:

Use Regularization to assign a finite value to this product:

The product of all natural numbers is divergent:

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

Many special functions are defined by products, including Factorial:

Pochhammer:

FactorialPower:

QPochhammer:

Hyperfactorial:

And BarnesG:

Many constants can be represented as products including Pi:

E:

Glaisher:

Weierstrass factorizations for elementary functions:

A Lagrange interpolating polynomial:

A Lagrange interpolating polynomial for symbolic values:

A Newton interpolating polynomial for symbolic values:

Using a symbolic closed‐form product can provide fast evaluation:

Compare timings of the closed form to procedural evaluation:

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: