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Wallis Integral

The Wallis integral is the common value

I_n=int_0^(pi/2)sin^nxdx=int_0^(pi/2)cos^nxdx,
(1)

where n is a real number greater than -1. In terms of the gamma function,

I_n=(sqrt(pi)Gamma((n+1)/2))/(2Gamma((n+2)/2)).
(2)

For n>1, integration by parts gives the recurrence equation

I_n=(n-1)/nI_(n-2),
(3)

with I_0=pi/2 and I_1=1. Consequently, for nonnegative integers,

I_n=((n-1)!!)/(n!!){pi/2 for even n; 1 for odd n,
(4)

where n!! is the double factorial. The recurrence equation and these special values are often called the Wallis sine and cosine formulas.

See also

Double Factorial, Gamma Function, Integration by Parts, Wallis Cosine Formula, Wallis Formula

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Wallis, J. Arithmetica Infinitorum. Oxford, England: Typis Leon. Lichfield, Impensis Tho. Robinson, 1656.

Cite this as:

Weisstein, Eric W. "Wallis Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WallisIntegral.html

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