In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product
infty | |
\prod | |
n=1 |
an=a1a2a3 …
is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true.
The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):
2 | |
\pi |
=
\sqrt{2 | |
}{ |
2} ⋅
\sqrt{2+\sqrt{2 | |
\pi | |
2 |
=\left(
2 | |
1 |
⋅
2 | |
3 |
\right) ⋅ \left(
4 | |
3 |
⋅
4 | |
5 |
\right) ⋅ \left(
6 | |
5 |
⋅
6 | |
7 |
\right) ⋅ \left(
8 | |
7 |
⋅
8 | |
9 |
\right) ⋅ … =
infty | |
\prod | |
n=1 |
\left(
4n2 | |
4n2-1 |
\right).
The product of positive real numbers
infty | |
\prod | |
n=1 |
an
infty | |
\sum | |
n=1 |
log(an)
If we define
an=1+pn
N | |
1+\sum | |
n=1 |
pn\le
N | |
\prod | |
n=1 |
\left(1+pn\right)\le\exp\left(
N | |
\sum | |
n=1 |
pn\right)
show that the infinite product of an converges if the infinite sum of the pn converges. This relies on the Monotone convergence theorem. We can show the converse by observing that, if
pn\to0
\limn
log(1+pn) | |
pn |
=\limx\to
log(1+x) | |
x |
=1,
and by the limit comparison test it follows that the two series
infty | |
\sum | |
n=1 |
log(1+pn) and
infty | |
\sum | |
n=1 |
pn,
are equivalent meaning that either they both converge or they both diverge.
If the series diverges to
-infty
For the case where the
pn
pn=
(-1)n | |
\sqrt{n |
See main article: Weierstrass factorization theorem.
One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, each with at most a single root. In general, if f has a root of order m at the origin and has other complex roots at u1, u2, u3, ... (listed with multiplicities equal to their orders), then
f(z)=zme\phi(z)
infty | |
\prod | |
n=1 |
\left(1-
z | |
un |
\right)\exp\left\lbrace
z | |
un |
+
1 | \left( | |
2 |
z | |
un |
\right)2+ … +
1 | \left( | |
λn |
z | |
un |
λn | |
\right) |
\right\rbrace
where λn are non-negative integers that can be chosen to make the product converge, and
\phi(z)
f(z)=zme\phi(z)
infty | |
\prod | |
n=1 |
\left(1-
z | |
un |
\right).
This can be regarded as a generalization of the fundamental theorem of algebra, since for polynomials, the product becomes finite and
\phi(z)
In addition to these examples, the following representations are of special note:
Function | Infinite product representation(s) | Notes | |||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Simple pole | \begin{align}
&=
\\ &=
\right) \end{align} | ||||||||||||||||||||||||||||||||||||
Sinc function |
\left(1-
\right) | This is due to Euler. Wallis' formula for π is a special case of this. | |||||||||||||||||||||||||||||||||||
Reciprocal gamma function | \begin{align}
&=ze\gamma
\left(1+
\right)
\\ &=z
\end{align} | Schlömilch | |||||||||||||||||||||||||||||||||||
Weierstrass sigma function | \sigma(z)=
| Here Λ* | |||||||||||||||||||||||||||||||||||
Q-Pochhammer symbol | (z;q)infty=
(1-zqn) | Widely used in q-analog theory. The Euler function is a special case. | |||||||||||||||||||||||||||||||||||
Ramanujan theta function | \begin{align} f(a,b)
\\ &=
(1+an+1bn)(1+anbn+1)(1-an+1bn+1) \end{align} | An expression of the Jacobi triple product, also used in the expression of the Jacobi theta function | |||||||||||||||||||||||||||||||||||
Riemann zeta function | \zeta(z)=
| Here pn denotes the nth prime number. This is a special case of the Euler product. |
The last of these is not a product representation of the same sort discussed above, as ζ is not entire. Rather, the above product representation of ζ(z) converges precisely for Re(z) > 1, where it is an analytic function. By techniques of analytic continuation, this function can be extended uniquely to an analytic function (still denoted ζ(z)) on the whole complex plane except at the point z = 1, where it has a simple pole.