“Gregory's series”的意思、由来-开放百科全书

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) {{harv|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.^[1]

The series

Compare with the series for sine, which is similar but has factorials in the denominator.

History

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),^[2]

Gregory is cited for the series based on two publications in 1668, Geometriae pars universalis (The Universal Part of Geometry), Exercitationes geometrica (Geometrical Exercises).

See also

References

1. ^{{cite web|title=Gregory Series|url=http://mathworld.wolfram.com/GregorySeries.html|publisher=Wolfram Math World|accessdate=26 July 2012}}
2. ^{{cite web|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf |title=Tantrasamgraha with English translation |editor=K.V. Sarma |others=Translated by V.S. Narasimhan |publisher=Indian National Academy of Science |pages=48 |language=Sanskrit, English |accessdate=17 January 2010 |deadurl=yes |archiveurl=https://web.archive.org/web/20120309014402/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf |archivedate=9 March 2012 |df= }}
3. ^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
4. ^{{cite web| editor=K. V. Sarma & S Hariharan| work=Yuktibhāṣā of Jyeṣṭhadeva|url=http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf| title=A book on rationales in Indian Mathematics and Astronomy—An analytic appraisal| accessdate=2006-07-09|format=PDF |archiveurl = https://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf |archivedate = 28 September 2006}}
5. ^{{Cite book|last=C.K. Raju|title=Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transsmision of the Calculus from India to Europe in the 16 c. CE|publisher=Centre for Studies in Civilistaion|location=New Delhi|year=2007|url=https://books.google.com/books/about/Cultural_Foundations_of_Mathematics.html?id=jza_cNJM6fAC|series=History of Science, Philosophy and Culture in Indian Civilisation|volume=X Part 4|page=231|isbn=81-317-0871-3}}