From Wikipedia, the free encyclopedia
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.
Thus
![{\displaystyle Q\left(\prod _{x}f(x)\right)=f(x)\,.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ce952a2e6377dd42c786e032223b6de4a2261aca)
More explicitly, if
, then
![{\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/c4bd048c2a7c2409eae7bace5f0a3654f2327f0a)
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.
If
is a period of function
then
![{\displaystyle \prod _{x}f(Tx)=Cf(Tx)^{x-1}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/19d1a01633768ca524c96fabd2942b82d5b9a579)
Connection to indefinite sum
[edit]
Indefinite product can be expressed in terms of indefinite sum:
![{\displaystyle \prod _{x}f(x)=\exp \left(\sum _{x}\ln f(x)\right)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/3a28ea6c042f7dc5a3f0fae3d20918695066ea0c)
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.
.
![{\displaystyle \prod _{x}f(x)g(x)=\prod _{x}f(x)\prod _{x}g(x)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/7db4c9172f6e35edd63fef030167ca35aca88549)
![{\displaystyle \prod _{x}f(x)^{a}=\left(\prod _{x}f(x)\right)^{a}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6355dfd6db8d99ea7882984d0455c4a15bd4724c)
![{\displaystyle \prod _{x}a^{f(x)}=a^{\sum _{x}f(x)}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/14de7e4f75eaa0d1f047bf2a03b9e4fe84d27015)
List of indefinite products
[edit]
This is a list of indefinite products
. Not all functions have an indefinite product which can be expressed in elementary functions.
![{\displaystyle \prod _{x}a=Ca^{x}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/49e4d01bc5718bc89a0bd7890b0abf398ee7503c)
![{\displaystyle \prod _{x}x=C\,\Gamma (x)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/40cd10914a99efde69a0004588c1ae11819865d0)
![{\displaystyle \prod _{x}{\frac {x+1}{x}}=Cx}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/0cefd655cb7bb2a0602249e28b687128be6b1f57)
![{\displaystyle \prod _{x}{\frac {x+a}{x}}={\frac {C\,\Gamma (x+a)}{\Gamma (x)}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/5f973a856a046c0143dee76504ff0da2b137fc30)
![{\displaystyle \prod _{x}x^{a}=C\,\Gamma (x)^{a}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6c4077c061383ced76e2786036d9998b5ad928d9)
![{\displaystyle \prod _{x}ax=Ca^{x}\Gamma (x)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/c22e4b6a8ddd741be62153658e76c3e9426eae90)
![{\displaystyle \prod _{x}a^{x}=Ca^{{\frac {x}{2}}(x-1)}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/405dfdde2d8df22e06169789f51f45ed13cbc8f8)
![{\displaystyle \prod _{x}a^{\frac {1}{x}}=Ca^{\frac {\Gamma '(x)}{\Gamma (x)}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/908b969b128405899e9c8a270687f6f00cbd04ec)
![{\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)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/983dc8904d520d9b7fd801d268092961bcbaaf35)
- (see K-function)
![{\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)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d2026b27170487c28a1cc432789bd03f789cc25a)
- (see Barnes G-function)
![{\displaystyle \prod _{x}\operatorname {sexp} _{a}(x)={\frac {C\,(\operatorname {sexp} _{a}(x))'}{\operatorname {sexp} _{a}(x)(\ln a)^{x}}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/96653c9e0007a81597445d2d6f0875053e4535cc)
- (see super-exponential function)
![{\displaystyle \prod _{x}x+a=C\,\Gamma (x+a)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ee1b7d760f354b95d006e8946b1e4c2bf0f3def8)
![{\displaystyle \prod _{x}ax+b=C\,a^{x}\Gamma \left(x+{\frac {b}{a}}\right)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/13fe365fe99f866f4b3617fe5630e73dd4705ca1)
![{\displaystyle \prod _{x}ax^{2}+bx=C\,a^{x}\Gamma (x)\Gamma \left(x+{\frac {b}{a}}\right)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/170079d5e58b9281c2ae7cabd0e728d7b0413dc9)
![{\displaystyle \prod _{x}x^{2}+1=C\,\Gamma (x-i)\Gamma (x+i)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/cc2da00cf69dfbc02f2812a43623ba2a5e11bbf8)
![{\displaystyle \prod _{x}x+{\frac {1}{x}}={\frac {C\,\Gamma (x-i)\Gamma (x+i)}{\Gamma (x)}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/c5d5f9614d4826598f49c22c414de3109544b319)
![{\displaystyle \prod _{x}\csc x\sin(x+1)=C\sin x}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/9f912e59ada79720b1ac94530e3018c1c3d48c04)
![{\displaystyle \prod _{x}\sec x\cos(x+1)=C\cos x}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/627bb9c767061c087e991da2d7faa1e398ff8311)
![{\displaystyle \prod _{x}\cot x\tan(x+1)=C\tan x}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6306f6ddcaddfa24aafaaa81215d20c0236ed551)
![{\displaystyle \prod _{x}\tan x\cot(x+1)=C\cot x}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/8a3386b5adfe6f24835eb347446056501d92fc9e)