## Wallis Formula

The Wallis formula follows from the Infinite Product representation of the Sine

 (1)

Taking gives
 (2)

so
 (3)

A derivation due to Y. L. Yung uses the Riemann Zeta Function. Define
 (4) (5)

so
 (6)

Taking the derivative of the zeta function expression gives
 (7)

 (8)
Equating and squaring then gives the Wallis formula, which can also be expressed
 (9)

The q-Analog of the Wallis formula for is

 (10)

(Finch).

See also Wallis Cosine Formula, Wallis Sine Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 258, 1972.

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

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 63-64, 1951.

© 1996-9 Eric W. Weisstein
1999-05-26