Wallis Formula

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

$$\sin x = x \prod_{n=1}^\infty \left({1 - {x^2\over \pi^2n^2}}\right).$$ (1)

Taking $x =\pi/2$ gives

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

so

$${\pi\over 2} = \prod_{n=1}^\infty \left[{(2n)^2\over(2n-1)(2... ...3}\,{4\cdot 4\over 3\cdot 5}\,{6\cdot 6\over 5\cdot 7} \cdots.$$ (3)

A derivation due to Y. L. Yung uses the Riemann Zeta Function. Define

$$F(s) \equiv -\mathop{\rm Li}\nolimits _s(-1)=\sum_{n=1}^\infty {(-1)^n\over n^s}$$

$$= (1-2^{1-s})\zeta(s)$$ (4)

$$F'(s) = \sum_{n=1}^\infty {(-1)^n\ln n\over n^s},$$ (5)

so

$$F'(0) = \sum_{n=1}^\infty (-1)^n\ln n=-\ln 1+\ln 2-\ln 3+\ldots$$

$$= \ln\left({2\cdot 4\cdot 6\cdots\over 1\cdot 3\cdot 5\cdots}\right).$$ (6)

Taking the derivative of the zeta function expression gives

$${d\over ds}(1-2^{1-s})\zeta(s)=2^{1-s}(\ln 2)\zeta(s)+(1-2^{1-s})\zeta'(s)$$ (7)

$\left[{{d\over ds}(1-2^{1-s})\zeta(s)}\right]_{s=0}=-\ln 2-\zeta'(0)$
$=-\ln 2+{\textstyle{1\over 2}}\ln(2\pi)=\ln\left({{\sqrt{2\pi}\over 2}\,}\right)=\ln\left({\sqrt{\pi\over 2}\,}\right).\quad$	(8)

Equating and squaring then gives the Wallis formula, which can also be expressed

$${\pi\over 2}=\left[{4^{\zeta(0)}e^{-\zeta'(0)}}\right]^2.$$ (9)

The q-Analog of the Wallis formula for $q=2$ is

$$\prod_{k=1}^\infty (1-q^{-k})^{-1}=3.4627466194\ldots$$ (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.