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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).
\end{displaymath} (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],
\end{displaymath} (2)

{\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.
\end{displaymath} (3)

A derivation due to Y. L. Yung uses the Riemann Zeta Function. Define
$\displaystyle F(s)$ $\textstyle \equiv$ $\displaystyle -\mathop{\rm Li}\nolimits _s(-1)=\sum_{n=1}^\infty {(-1)^n\over n^s}$  
  $\textstyle =$ $\displaystyle (1-2^{1-s})\zeta(s)$ (4)
$\displaystyle F'(s)$ $\textstyle =$ $\displaystyle \sum_{n=1}^\infty {(-1)^n\ln n\over n^s},$ (5)

$\displaystyle F'(0)$ $\textstyle =$ $\displaystyle \sum_{n=1}^\infty (-1)^n\ln n=-\ln 1+\ln 2-\ln 3+\ldots$  
  $\textstyle =$ $\displaystyle \ln\left({2\cdot 4\cdot 6\cdots\over 1\cdot 3\cdot 5\cdots}\right).$ (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)
\end{displaymath} (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.
\end{displaymath} (9)

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

\prod_{k=1}^\infty (1-q^{-k})^{-1}=3.4627466194\ldots
\end{displaymath} (10)


See also Wallis Cosine Formula, Wallis Sine Formula


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.''

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

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