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Machin's Formula

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Machin's formula is given by

1/4pi=4cot^(-1)5-cot^(-1)239.
(1)

For positive t, cot^(-1)t=tan^(-1)(1/t), so this is also an inverse tangent identity. Following the exponential-product calculation of Trott (2006), take a=tan^(-1)(1/5), b=tan^(-1)(1/239), and theta=4a-b. The Euler formula gives

e^(2itheta)=((5+i)/(5-i))^4(239-i)/(239+i)
(2)
=i.
(3)

This determines theta only modulo pi. Since 0<b<a<1/5, one has 0<theta<4/5<pi/2, which fixes the branch by selecting theta=pi/4.

There are a whole class of Machin-like formulas with various numbers of terms (although only four such formulas with only two terms). The properties of these formulas are intimately connected with cotangent identities.

See also

239, Gregory Number, Inverse Cotangent, Inverse Tangent, Machin-Like Formulas, Pi Formulas

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References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 105, 2003.Gourdon, X. and Sebah, P. "Collection of Series for pi." http://numbers.computation.free.fr/Constants/Pi/piSeries.html.Trott, M. "Operations on Trigonometric Expressions." §1.4 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Machin's Formula

Cite this as:

Weisstein, Eric W. "Machin's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MachinsFormula.html

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