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Wallis Formula

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The Wallis formula follows from the infinite product representation of the sine

sinx=xproduct_(n=1)^infty(1-(x^2)/(pi^2n^2)).
(1)

Taking x=pi/2 gives

1=pi/2product_(n=1)^infty[1-1/((2n)^2)]=pi/2product_(n=1)^infty[((2n)^2-1)/((2n)^2)],
(2)

so

pi/2=product_(n=1)^(infty)[((2n)^2)/((2n-1)(2n+1))]
(3)
=(2·2)/(1·3)(4·4)/(3·5)(6·6)/(5·7)...
(4)

(OEIS A052928 and A063196).

An accelerated product is given by

pi/2=e^s
(5)
=(2/1)^(1/2)((2^2)/(1·3))^(1/4)((2^3·4)/(1·3^3))^(1/8)((2^4·4^4)/(1·3^6·5))^(1/16)...
(6)

where

s=sum_(n=1)^infty1/(2^n)sum_(k=0)^n(-1)^(k+1)(n; k)ln(k+1)
(7)

(Guillera and Sondow 2005, Sondow 2005). This is analogous to the products

e^gamma=(2/1)^(1/2)((2^2)/(1·3))^(1/3)((2^3·4)/(1·3^3))^(1/4)((2^4·4^4)/(1·3^6·5))^(1/5)...
(8)

and

e=(2/1)^(1/1)((2^2)/(1·3))^(1/2)((2^3·4)/(1·3^3))^(1/3)((2^4·4^4)/(1·3^6·5))^(1/4)...
(9)

(Sondow 2005).

A derivation of equation (◇) due to Y. L. Yung (pers. comm., 1996; modified by J. Sondow, pers. comm., 2002) defines

F(s)=-Li_s(-1)
(10)
=1/2+1/2sum_(n=1)^(infty)(-1)^(n-1)[n^(-s)-(n+1)^(-s)]
(11)
=(1-2^(1-s))zeta(s),
(12)

where Li_s(x) is a polylogarithm and zeta(n) is the Riemann zeta function, which converges for R[s]>-1. Taking the derivative of (11) gives

F^'(s)=1/2sum_(n=1)^infty(-1)^n[(lnn)/(n^s)-(ln(n+1))/((n+1)^s)],
(13)

which also converges for R[s]>-1, and plugging in s=0 then gives

F^'(0)=1/2sum_(n=1)^(infty)(-1)^n[lnn-ln(n+1)]
(14)
=1/2[(-ln1+ln2)+(ln2-ln3)+(-ln3+ln4)+...]
(15)
=1/2ln((2·2)/(1·3)(4·4)/(3·5)(6·6)/(5·7)...).
(16)

Now, taking the derivative of the zeta function expression (◇) gives

d/(ds)(1-2^(1-s))zeta(s)=2^(1-s)(ln2)zeta(s)+(1-2^(1-s))zeta^'(s),
(17)

and again setting s=0 yields

F^'(0)=[d/(ds)(1-2^(1-s))zeta(s)]_(s=0)
(18)
=-ln2-zeta^'(0)
(19)
=-ln2+1/2ln(2pi)
(20)
=1/2ln(1/2pi),
(21)

where

zeta^'(0)=-1/2ln(2pi)=-0.918938...
(22)

(OEIS A075700) follows from the Hadamard product for the Riemann zeta function. Equating and squaring (◇) and (◇) then gives the Wallis formula.

This derivation of the Wallis formula from zeta^'(0) using the Hadamard product can also be reversed to derive zeta^'(0) from the Wallis formula without using the Hadamard product (Sondow 1994).

The Wallis formula can also be expressed as

pi/2=[4^(zeta(0))e^(-zeta^'(0))]^2.
(23)

The q-analog of the Wallis formula with q=1/2 is

product_(k=1)^(infty)(1-q^k)^(-1)=1/((1/2;1/2)_infty)
(24)
=3.4627466194...
(25)

(OEIS A065446; Finch 2003), where (q;a)_infty is the q-Pochhammer symbol. This constant is 1/Q, where Q is the constant encountered in digital tree searching. The form of the product is exactly the generating function for the partition function P due to Euler, and is related to q-pi.

See also

Pi Formulas, Pippenger Product, q-pi, Wallis Cosine Formula

Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 258, 1972.Finch, S. R. "Archimedes' Constant." §1.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 17-28, 2003.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005. http://arxiv.org/abs/math.NT/0506319.Jeffreys, H. and Jeffreys, B. S. "Wallis's Formula for pi." §15.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 468, 1988.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 63-64, 1951.Sloane, N. J. A. Sequences A052928, A063196, A065446, and A075700 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." Proc. Amer. Math. Soc. 120, 421-424, 1994.Sondow, J. "A Faster Product for pi and a New Integral for ln(pi/2)." Amer. Math. Monthly 112, 729-734, 2005.Wallis, J. Arithmetica Infinitorum. Oxford, England, 1656.

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Wallis Formula

Cite this as:

Sondow, Jonathan and Weisstein, Eric W. "Wallis Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WallisFormula.html

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