## 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 , the Product converges to a Nonzero number Iff converges.

Infinite products can be used to define the Cosine

 (1)

Gamma Function
 (2)

Sine, and Sinc Function. They also appear in the Polygon Circumscribing Constant
 (3)

An interesting infinite product formula due to Euler which relates and the th Prime is
 (4) (5)

(Blatner 1997).

The product

 (6)

has closed form expressions for small Positive integral ,
 (7) (8) (9) (10)

The d-Analog expression
 (11)

also has closed form expressions,
 (12) (13) (14) (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.