Indefinite product

In mathematics, the indefinite product operator is the inverse operator of $Q(f(x)) = \frac$. It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors use term discrete multiplicative integration.

Thus

:$Q\left( \prod_x f(x) \right) = f(x) \, .$

More explicitly, if $\prod_x f(x) = F(x)$, then

:$\frac = f(x) \, .$

If ''F''(''x'') is a solution of this functional equation for a given ''f''(''x''), then so is ''CF''(''x'') for any constant ''C''. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant.

Period rule

If $T$ is a period of function $f(x)$ then

:$\prod _x f(Tx)=C f(Tx)^$

Connection to indefinite sum

Indefinite product can be expressed in terms of indefinite sum:

:$\prod _x f(x)= \exp \left(\sum _x \ln f(x)\right)$

Alternative usage

Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which the numerical value of the upper limit is not given.Algorithms for Nonlinear Higher Order Difference Equations
Manuel Kauers e.g.

:$\prod_^n f(k)$.

Rules

:$\prod _x f(x)g(x) = \prod _x f(x)\prod _x g(x)$

:$\prod _x f(x)^a = \left(\prod _x f(x)\right)^a$

:$\prod _x a^ = a^$

List of indefinite products

This is a list of indefinite products $\prod _x f(x)$. Not all functions have an indefinite product which can be expressed in elementary functions.

:$\prod _x a = C a^x$

:$\prod _x x = C\, \Gamma (x)$

:$\prod _x \frac = C x$

:$\prod _x \frac = \frac$

:$\prod _x x^a = C\, \Gamma (x)^a$

:$\prod _x ax = C a^x \Gamma (x)$

:$\prod _x a^x = C a^$

:$\prod _x a^ = C a^$

:$\prod _x x^x= C\, e^= C\,e^= C\, \operatorname(x)$

:(see K-function)

:$\prod _x \Gamma(x) = \frac = C\,\Gamma(x)^ e^= C\, \operatorname(x)$

:(see Barnes G-function)

:$\prod _x \operatorname_a(x) = \frac$

:(see super-exponential function)

:$\prod _x x+a = C\,\Gamma (x+a)$

:$\prod _x ax+b = C\, a^x \Gamma \left(x+\frac\right)$

:$\prod _x ax^2+bx = C\,a^x \Gamma (x) \Gamma \left(x+\frac\right)$

:$\prod _x x^2+1 = C\, \Gamma (x-i) \Gamma (x+i)$

:$\prod _x x+\frac = \frac$

:$\prod _x \csc x \sin (x+1) = C \sin x$

:$\prod _x \sec x \cos (x+1) = C \cos x$

:$\prod _x \cot x \tan (x+1) = C \tan x$

:$\prod _x \tan x \cot (x+1) = C \cot x$

See also

* Indefinite sum
* Product integral
*List of derivatives and integrals in alternative calculi
* Fractal derivative

References

Further reading

* http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica


- bug in Maple V to Maple 8 handling of indefinite product

Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations

Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities

External links

Non-Newtonian calculus website

{{DEFAULTSORT:Indefinite Product
Mathematical analysis
Indefinite sums
Indefinite sums
Non-Newtonian calculus