Wallis product

In mathematics, Wallis' product for π, written down in 1655 by John Wallis, states that

Wallis derived this infinite product as it is done in calculus books today, by examining ${\displaystyle \textstyle \int _{0}^{\pi }\sin ^{n}x\,dx}$ for even and odd values of n, and noting that for large n, increasing n by 1 results in a change that becomes ever smaller as n increases. Since modern infinitesimal calculus did not yet exist then, and the mathematical analysis of the time was inadequate to discuss the convergence issues, this was a hard piece of research, and tentative as well.

Wallis' product is, in retrospect, an easy corollary of the later Euler formula for the sine function. In 2015 researchers C. R. Hagen and Tamar Friedmann, in a surprise discovery, found the same formula in quantum mechanical calculations of the energy levels of a hydrogen atom.

Let x = π/2:

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