In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:[1]
This series converges in the complex disk except for (where ).
It was first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma (c. 1340 – c. 1425), the founder of the Kerala school, and is described in extant works by Nīlakaṇṭha Somayāji (c. 1500) and Jyeṣṭhadeva (c. 1530). Mādhava's work was unknown in Europe, and the arctangent series was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673.[2] In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series).[3]
The special case of the arctangent of is traditionally called the Leibniz formula for π, or recently sometimes the Mādhava–Leibniz formula:
The extremely slow convergence of the arctangent series for makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for . Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.
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^Roy 1990.
^For example: Gupta 1973, Gupta 1987; Joseph, George Gheverghese (2011) [1st ed. 1991]. The Crest of the Peacock: Non-European Roots of Mathematics (3rd ed.). Princeton University Press. p. 428. Levrie, Paul (2011). "Lost and Found: An Unpublished ζ(2)-Proof". Mathematical Intelligencer. 33: 29–32. doi:10.1007/s00283-010-9179-y. S2CID 121133743. Other combinations of names include, Madhava–Gregory–Leibniz series: Benko, David; Molokach, John (2013). "The Basel Problem as a Rearrangement of Series". College Mathematics Journal. 44 (3): 171–176. doi:10.4169/college.math.j.44.3.171. S2CID 124737638. Madhava–Leibniz–Gregory series: Danesi, Marcel (2021). "1. Discovery of π and Its Manifestations". Pi (π) in Nature, Art, and Culture. Brill. pp. 1–30. doi:10.1163/9789004433397_002. ISBN 978-90-04-43337-3. S2CID 242107102. Nilakantha–Gregory series: Campbell, Paul J. (2004). "Borwein, Jonathan, and David Bailey, Mathematics by Experiment". Reviews. Mathematics Magazine. 77 (2): 163. doi:10.1080/0025570X.2004.11953245. S2CID 218541218. Gregory–Leibniz–Nilakantha formula: Gawrońska, Natalia; Słota, Damian; Wituła, Roman; Zielonka, Adam (2013). "Some generalizations of Gregory's power series and their applications" (PDF). Journal of Applied Mathematics and Computational Mechanics. 12 (3): 79–91. doi:10.17512/jamcm.2013.3.09.
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