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 Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are:
All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669, and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673, and is conventionally called Gregory's series. The specific value can be used to calculate the circle constant pi, 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, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations).
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.
None of Madhava's works, containing any of the series expressions attributed to him, have survived. These series expressions are found in the writings of the followers of Madhava in the Kerala school. At many places these authors have clearly stated that these are "as told by Madhava". Thus the enunciations of the various series found in Tantrasamgraha and its commentaries can be safely assumed to be in "Madhava's own words". The translations of the relevant verses as given in the Yuktidipika commentary of Tantrasamgraha (also known as Tantrasamgraha-vyakhya) by Sankara Variar (circa. 1500 - 1560 CE) are reproduced below.
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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 Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are: All three series were later independently discovered in 17th century Europe.
Yuktibhasa (en malayalam : യുക്തിഭാഷ) ou Ganita Yuktibhasa est un traité de mathématiques et d'astronomie, écrit par l'astronome indien , membre de l'école du Kerala, en 1530. Le traité récapitule les travaux de Madhava de Sangamagrama, Nilakantha Somayaji, Parameswara, Jyeṣṭhadeva, et d'autres astronomes mathématiciens de cette école. Plusieurs historiens voient dans le Yuktibhasa le premier traité d'analyse, devançant de trois siècles la redécouverte du calcul infinitésimal par les occidentaux.
Les fonctions circulaires réciproques, ou fonctions trigonométriques inverses, sont les fonctions réciproques des fonctions circulaires, pour des intervalles de définition précis. Les fonctions réciproques des fonctions sinus, cosinus, tangente, cotangente, sécante et cosécante sont appelées arc sinus, arc cosinus, arc tangente, arc cotangente, arc sécante et arc cosécante. Les fonctions circulaires réciproques servent à obtenir un angle à partir de l'une quelconque de ses lignes trigonométriques, mais aussi à expliciter les primitives de certaines fonctions.
Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond
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