Infinite Product

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

A Product involving an Infinite number of terms. Such products can converge. In fact, for Positive $a_n$, the Product $\prod_{n=1}^\infty a_n$ converges to a Nonzero number Iff $\sum_{n=1}^\infty \ln a_n$ converges.

Infinite products can be used to define the Cosine

$$\cos x=\prod_{n=1}^\infty \left[{1-{4x^2\over\pi^2(2n-1)^2}}\right],$$ (1)

Gamma Function

$$\Gamma(z)=\left[{ze^{\gamma z}\prod_{r=1}^\infty \left({1 + {z\over r}}\right)e^{-z/r}}\right]^{-1},$$ (2)

Sine, and Sinc Function. They also appear in the Polygon Circumscribing Constant

$$K= \prod_{n=3}^\infty {1\over\cos\left({\pi\over n}\right)}.$$ (3)

An interesting infinite product formula due to Euler which relates $\pi$ and the $n$th Prime $p_n$ is

$$\pi = {2\over \prod_{i=n}^\infty \left[{1+{\sin({\textstyle{1\over 2}}\pi p_n)\over p_n}}\right]}$$ (4)

$$= {2\over \prod_{i=n}^\infty \left[{1+{(-1)^{(p_n-1)/2}\over p_n}}\right]}$$ (5)

(Blatner 1997).

The product

$$\prod_{n=1}^\infty \left({1+{1\over n^p}}\right)$$ (6)

has closed form expressions for small Positive integral $p\geq 2$,

$$\prod_{n=1}^\infty\left({1+{1\over n^2}}\right) = {\sinh\pi\over\pi}$$ (7)

$$\prod_{n=1}^\infty\left({1+{1\over n^3}}\right) = {1\over\pi}\cosh({\textstyle{1\over 2}}\pi\sqrt{3}\,)$$ (8)

$$\prod_{n=1}^\infty\left({1+{1\over n^4}}\right) = {\cosh(\pi\sqrt{2}\,)-\cos(\pi\sqrt{2}\,)\over 2\pi^2}$$ (9)

$$\prod_{n=1}^\infty\left({1+{1\over n^5}}\right) = \left\vert{\Gamma[\mathop{\rm exp}\nolimits ({\textstyle{2\over 5... ...amma[\mathop{\rm exp}\nolimits ({\textstyle{6\over 5}}\pi i)]}\right\vert^{-2}.$$ (10)

The d-Analog expression

$$[\infty!]_d=\prod_{n=3}^\infty \left({1-{2^d\over n^d}}\right)$$ (11)

also has closed form expressions,

$$\prod_{n=3}^\infty\left({1-{4\over n^2}}\right) = {1\over 6}$$ (12)

$$\prod_{n=3}^\infty\left({1-{8\over n^3}}\right) = {\sinh(\pi\sqrt{3}\,)\over 42\pi\sqrt{3}}$$ (13)

$$\prod_{n=3}^\infty\left({1-{16\over n^4}}\right) = {\sinh(2\pi)\over 120\pi}$$ (14)

$$\prod_{n=3}^\infty\left({1-{32\over n^5}}\right) = \left\vert{\Gamma[\mathop{\rm exp}\nolimits ({\textstyle{1\over 5... ...mma[2\mathop{\rm exp}\nolimits ({\textstyle{7\over 5}}\pi i)]}\right\vert^{-2}.$$ (15)

See also Cosine, Dirichlet Eta Function, Euler Identity, Gamma Function, Iterated Exponential Constants, Polygon Circumscribing Constant, Polygon Inscribing Constant, Q-Function, Sine

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 75, 1972.

Arfken, G. ``Infinite Products.'' §5.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 346-351, 1985.

Blatner, D. The Joy of Pi. New York: Walker, p. 119, 1997.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/infprd/infprd.html

Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.

Whittaker, E. T. and Watson, G. N. §7.5 and 7.6 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.