In mathematics, the indefinite product operator is the inverse operator of . 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.[citation needed]
Thus
More explicitly, if , then
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.
Indefinite product can be expressed in terms of indefinite sum:
Alternative usageedit
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.[1] e.g.
.
Rulesedit
List of indefinite productsedit
This is a list of indefinite products . Not all functions have an indefinite product which can be expressed in elementary functions.
[1] - bug in Maple V to Maple 8 handling of indefinite product
Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
External linksedit
Non-Newtonian calculus website
December 18, 2023
indefinite, product, mathematics, indefinite, product, operator, inverse, operator, textstyle, frac, discrete, version, geometric, integral, geometric, calculus, newtonian, calculi, some, authors, term, discrete, multiplicative, integration, citation, needed, . In mathematics the indefinite product operator is the inverse operator of Q f x f x 1 f x textstyle Q f x frac f x 1 f x 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 citation needed Thus Q x f x f x displaystyle Q left prod x f x right f x More explicitly if x f x F x textstyle prod x f x F x then F x 1 F x f x displaystyle frac F x 1 F x 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 Contents 1 Period rule 2 Connection to indefinite sum 3 Alternative usage 4 Rules 5 List of indefinite products 6 See also 7 References 8 Further reading 9 External linksPeriod rule editIf T displaystyle T nbsp is a period of function f x displaystyle f x nbsp then x f T x C f T x x 1 displaystyle prod x f Tx Cf Tx x 1 nbsp Connection to indefinite sum editIndefinite product can be expressed in terms of indefinite sum x f x exp x ln f x displaystyle prod x f x exp left sum x ln f x right nbsp Alternative usage editSome 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 1 e g k 1 n f k displaystyle prod k 1 n f k nbsp Rules edit x f x g x x f x x g x displaystyle prod x f x g x prod x f x prod x g x nbsp x f x a x f x a displaystyle prod x f x a left prod x f x right a nbsp x a f x a x f x displaystyle prod x a f x a sum x f x nbsp List of indefinite products editThis is a list of indefinite products x f x textstyle prod x f x nbsp Not all functions have an indefinite product which can be expressed in elementary functions x a C a x displaystyle prod x a Ca x nbsp x x C G x displaystyle prod x x C Gamma x nbsp x x 1 x C x displaystyle prod x frac x 1 x Cx nbsp x x a x C G x a G x displaystyle prod x frac x a x frac C Gamma x a Gamma x nbsp x x a C G x a displaystyle prod x x a C Gamma x a nbsp x a x C a x G x displaystyle prod x ax Ca x Gamma x nbsp x a x C a x 2 x 1 displaystyle prod x a x Ca frac x 2 x 1 nbsp x a 1 x C a G x G x displaystyle prod x a frac 1 x Ca frac Gamma x Gamma x nbsp x x x C e z 1 x z 1 C e ps 2 z z 2 z 2 z 2 ln 2 p C K x displaystyle prod x x x C e zeta prime 1 x zeta prime 1 C e psi 2 z frac z 2 z 2 frac z 2 ln 2 pi C operatorname K x nbsp see K function x G x C G x x 1 K x C G x x 1 e z 2 ln 2 p z 2 z 2 ps 2 z C G x displaystyle prod x Gamma x frac C Gamma x x 1 operatorname K x C Gamma x x 1 e frac z 2 ln 2 pi frac z 2 z 2 psi 2 z C operatorname G x nbsp see Barnes G function x sexp a x C sexp a x sexp a x ln a x displaystyle prod x operatorname sexp a x frac C operatorname sexp a x operatorname sexp a x ln a x nbsp see super exponential function x x a C G x a displaystyle prod x x a C Gamma x a nbsp x a x b C a x G x b a displaystyle prod x ax b C a x Gamma left x frac b a right nbsp x a x 2 b x C a x G x G x b a displaystyle prod x ax 2 bx C a x Gamma x Gamma left x frac b a right nbsp x x 2 1 C G x i G x i displaystyle prod x x 2 1 C Gamma x i Gamma x i nbsp x x 1 x C G x i G x i G x displaystyle prod x x frac 1 x frac C Gamma x i Gamma x i Gamma x nbsp x csc x sin x 1 C sin x displaystyle prod x csc x sin x 1 C sin x nbsp x sec x cos x 1 C cos x displaystyle prod x sec x cos x 1 C cos x nbsp x cot x tan x 1 C tan x displaystyle prod x cot x tan x 1 C tan x nbsp x tan x cot x 1 C cot x displaystyle prod x tan x cot x 1 C cot x nbsp See also editIndefinite sum Product integral List of derivatives and integrals in alternative calculi Fractal derivativeReferences edit Algorithms for Nonlinear Higher Order Difference Equations Manuel KauersFurther reading edithttp reference wolfram com mathematica ref Product html Indefinite products with Mathematica 1 bug in Maple V to Maple 8 handling of indefinite product Markus Muller 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 IdentitiesExternal links editNon Newtonian calculus website Retrieved from https en wikipedia org w index php title Indefinite product amp oldid 1087213373, wikipedia, wiki, book, books, library,