Madhava series information

In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics.^[1] Using modern notation, these series are:

{\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}\theta ^{2k+1},\\[10mu]\cos \theta &=1-{\frac {\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}\theta ^{2k},\\[10mu]\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}|x|\leq 1.\end{alignedat}}

All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669,^[2] and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673,^[3] and is conventionally called Gregory's series. The specific value ${\textstyle \arctan 1={\tfrac {1}{4}}\pi }$ can be used to calculate the circle constant $π$ , and the arctangent series for $1$ is conventionally called Leibniz's series.

In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series,^[4] Madhava–Gregory series,^[5] or Madhava–Leibniz series^[6] (among other combinations).^[7]

No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later Kerala school mathematicians Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series.

^ Gupta 1987; Katz 1995; Roy 2021, Ch. 1. Power Series in Fifteenth-Century Kerala, pp. 1–22
^ Newton (1669) De analysi per aequationes numero terminorum infinitas was circulated as a manuscript but not published until 1711. For context, see:
Roy 2021, Ch. 8. De Analysi per Aequationes Infinitas, pp. 165–185.
Leibniz later included the series for sine and cosine in Leibniz (1676) De quadratura arithmetica circuli ellipseos et hyperbola cujus corollarium est trigonometria sine tabulis, which was only finally published in 1993. However, he had been sent Newton's sine and cosine series by Henry Oldenburg in 1675 and did not claim to have discovered them. See:
Probst, Siegmund (2015). "Leibniz as reader and second inventor: The cases of Barrow and Mengoli". In Goethe, N.; Beeley, P.; Rabouin, D. (eds.). G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes. Vol. 41. Springer. pp. 111–134. doi:10.1007/978-94-017-9664-4_6. ISBN 978-94-017-9663-7.
^ Gregory received a letter from John Collins including Newton's sine and cosine series in late 1670. He discovered the general Taylor series and sent a now-famous letter back to Collins in 1671 including several specific series including the arctangent. See Roy 1990.
Horvath, Miklos (1983). "On the Leibnizian quadrature of the circle" (PDF). Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica). 4: 75–83.
^ For example:
Plofker, Kim (2005). "Relations between approximations to the sine in Kerala mathematics". In Emch, Gérard G.; Sridharan, R.; Srinivas, M. D. (eds.). Contributions to the History of Indian Mathematics. Gurgaon: Hindustan Book Agency. pp. 135–152. doi:10.1007/978-93-86279-25-5_6. ISBN 978-81-85931-58-6.
Filali, Mahmoud (2012). "Harmonic analysis and applications". Kybernetes. 41: 129–144. doi:10.1108/03684921211213160. S2CID 206377839.
^ For example: Gupta 1973; Joseph 2011, 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.
^ For example: Gupta 1992;
Pouvreau, David (2015). "Sur l'accélération de la convergence de la série de Madhava-Leibniz". Quadrature (in French). 97: 17–25.
Young, Paul Thomas (2022). "From Madhava–Leibniz to Lehmer's Limit". American Mathematical Monthly. 129 (6): 524–538. doi:10.1080/00029890.2022.2051405. S2CID 247982859.
^ For example,
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.

[1] Gupta 1987; Katz 1995; Roy 2021, Ch. 1. Power Series in Fifteenth-Century Kerala, pp. 1–22

[2] Newton (1669) De analysi per aequationes numero terminorum infinitas was circulated as a manuscript but not published until 1711. For context, see:
Roy 2021, Ch. 8. De Analysi per Aequationes Infinitas, pp. 165–185.
Leibniz later included the series for sine and cosine in Leibniz (1676) De quadratura arithmetica circuli ellipseos et hyperbola cujus corollarium est trigonometria sine tabulis, which was only finally published in 1993. However, he had been sent Newton's sine and cosine series by Henry Oldenburg in 1675 and did not claim to have discovered them. See:
Probst, Siegmund (2015). "Leibniz as reader and second inventor: The cases of Barrow and Mengoli". In Goethe, N.; Beeley, P.; Rabouin, D. (eds.). G.W. Leibniz, Interrelations between Mathematics and Philosophy. Archimedes. Vol. 41. Springer. pp. 111–134. doi:10.1007/978-94-017-9664-4_6. ISBN 978-94-017-9663-7.

[3] Gregory received a letter from John Collins including Newton's sine and cosine series in late 1670. He discovered the general Taylor series and sent a now-famous letter back to Collins in 1671 including several specific series including the arctangent. See Roy 1990.
Horvath, Miklos (1983). "On the Leibnizian quadrature of the circle" (PDF). Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica). 4: 75–83.

[4] For example:
Plofker, Kim (2005). "Relations between approximations to the sine in Kerala mathematics". In Emch, Gérard G.; Sridharan, R.; Srinivas, M. D. (eds.). Contributions to the History of Indian Mathematics. Gurgaon: Hindustan Book Agency. pp. 135–152. doi:10.1007/978-93-86279-25-5_6. ISBN 978-81-85931-58-6.
Filali, Mahmoud (2012). "Harmonic analysis and applications". Kybernetes. 41: 129–144. doi:10.1108/03684921211213160. S2CID 206377839.

[5] For example: Gupta 1973; Joseph 2011, 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.

[6] For example: Gupta 1992;
Pouvreau, David (2015). "Sur l'accélération de la convergence de la série de Madhava-Leibniz". Quadrature (in French). 97: 17–25.
Young, Paul Thomas (2022). "From Madhava–Leibniz to Lehmer's Limit". American Mathematical Monthly. 129 (6): 524–538. doi:10.1080/00029890.2022.2051405. S2CID 247982859.

[7] For example,
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.

Madhava series information

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