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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 (4) 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 (12) 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 (16) and (21) 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. https://arxiv.org/abs/math/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: Typis Leon. Lichfield, Impensis Tho. Robinson, 1656.

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Wallis Formula

Cite this as:

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

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