# What is Wallis formula

Geometry > Trigonometry > Trigonometric Identities >

MathWorld Contributors > Somos >

MathWorld Contributors > Sondow >

Interactive Entries > Interactive Demonstrations >

## Wallis Formula

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

(1) |

Taking gives

(2) |

so

(OEIS A052928 and A063196).

An accelerated product is given by

where

(7) |

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

(8) |

and

(9) |

(Sondow 2005).

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

where is a polylogarithm and is the Riemann zeta function, which converges for . Taking the derivative of (11) gives

(13) |

which also converges for , and plugging in then gives

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

(17) |

and again setting yields

where

(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 using the Hadamard product can also be reversed to derive from the Wallis formula without using the Hadamard product (Sondow 1994).

The Wallis formula can also be expressed as

(23) |

The *q*-analog of the Wallis formula with is

(OEIS A065446; Finch 2003), where is the *q*-Pochhammer symbol. This constant is , where 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.

*q*-pi, Wallis Cosine Formula

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

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 ." §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 and a New Integral for ." *Amer. Math. Monthly***112**, 729-734, 2005.

Wallis, J. *Arithmetica Infinitorum.* Oxford, England, 1656.

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

## Wolfram Web Resources

Mathematica » The #1 tool for creating Demonstrations and anything technical. | Wolfram|Alpha » Explore anything with the first computational knowledge engine. | Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. |

Computerbasedmath.org » Join the initiative for modernizing math education. | Online Integral Calculator » Solve integrals with Wolfram|Alpha. | Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. |

Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. | Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. | Wolfram Language » Knowledge-based programming for everyone. |

- Whats a good code
- What is an immigration essay
- Can I play celtic harp
- Why are twins better than sisters
- What are the effects of fiscal policy
- How did Fukushima become so radioactive
- How do you get 25 000 Swagbucks
- Why do people not believe police brutality
- Is HCV sufficient for the JEE Mains
- What is the history of breakdancing
- Why is Express preferred over Node
- How can you evolve Riolu into Lucario
- How can I get scam emails
- What are the areas of math subjects
- What does the phrase choices chambers mean
- How did Kickstarter hack its initial growth
- Are time slips real according to physics
- Does nature ran out of imagination
- How do I log in to Gmail