Gregory's series, also known as the Madhava–Gregory series or Leibniz's series, is an infinite Taylor series expansion of the inverse tangent function. It was discovered by the Indian mathematician Madhava of Sangamagrama (1350–1410) (Gupta 1973) and rediscovered in 1668 by James Gregory. It was re-rediscovered a few years later by Gottfried Leibniz, who reobtained the Leibniz formula for π (which had also been obtained earlier by Madhava) as the special case x = 1 of the Gregory series.
The series is,
Compare with the series for sine, which is similar but has factorials in the denominator.
The earliest person to whom the series can be attributed with confidence is Madhava of Sangamagrama (c. 1340 – c. 1425). The original reference (as with much of Madhava's work) is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for arctanθ include Nilakantha Somayaji's Tantrasangraha (c. 1500),
Gregory is cited for the series based on two publications in 1668, Geometriae pars universalis (The Universal Part of Geometry), Exercitationes geometrica (Geometrical Exercises).
- "Gregory Series". Wolfram Math World. Retrieved 26 July 2012.
- K.V. Sarma (ed.). "Tantrasamgraha with English translation" (PDF) (in Sanskrit and English). Translated by V.S. Narasimhan. Indian National Academy of Science. p. 48. Archived from the original (PDF) on 9 March 2012. Retrieved 17 January 2010.
- Tantrasamgraha, ed. K.V. Sarma, trans. V. S. Narasimhan in the Indian Journal of History of Science, issue starting Vol. 33, No. 1 of March 1998
- K. V. Sarma & S Hariharan (ed.). "A book on rationales in Indian Mathematics and Astronomy—An analytic appraisal" (PDF). Yuktibhāṣā of Jyeṣṭhadeva. Archived from the original (PDF) on 28 September 2006. Retrieved 2006-07-09.
- C.K. Raju (2007). Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transsmision of the Calculus from India to Europe in the 16 c. CE. History of Science, Philosophy and Culture in Indian Civilisation. X Part 4. New Delhi: Centre for Studies in Civilistaion. p. 231. ISBN 81-317-0871-3.
- Carl B. Boyer, A history of mathematics, 2nd edition, by John Wiley & Sons, Inc., page 386, 1991
- Gupta, RC (1973). "The Madhava–Gregory series". Mathematical Education. 7: 67–70.