Madhava series: Difference between revisions

In [[mathematics]], a  '''Madhava series''' or '''Leibniz series''' is any one of the series in a collection of [[infinite series]] expressions all of which are believed to have been discovered by [[Madhava of Sangamagrama]] (c.&nbsp;1350 – c.&nbsp;1425), the founder of the [[Kerala school of astronomy and mathematics]] and later by [[Gottfried Wilhelm Leibniz]], among others. These expressions are the [[Maclaurin series]] expansions of the trigonometric [[sine]], [[cosine]] and [[arctangent]] [[Function (mathematics)|functions]], and the special case of the power series expansion of the arctangent function yielding a formula for computing π. The power series expansions of sine and cosine functions are respectively called ''Madhava's sine series'' and ''Madhava's cosine series''.  The power series expansion of the arctangent function is sometimes called ''Madhava–Gregory series''<ref>Reference to Gregory–Madhava series: {{Cite web|url=http://www.luigigobbi.com/EarliestKnownUsesOfSomeOfTheWordsOfMathematics/G-K.htm|title=Earliest Known Uses of Some of the Words of Mathematics|access-date=11 February 2010}}</ref><ref>Reference to Gregory–Madhava series: {{Cite web|url=http://www.mat.uc.pt/~jaimecs/pessoal/hpm.html|title=History of Mathematics in the classroom|last=Jaime Carvalho e Silva|date = July 1994|access-date=15 February 2010}}</ref> or ''Gregory–Madhava series''. These power series are also collectively called  ''Taylor–Madhava series''.<ref>{{Cite journal|title=Topic entry on complex analysis : Introduction|publisher=PlanetMath.org|url=https://planetmath.org/TopicEntryOnComplexAnalysis|access-date=10 February 2010}}</ref> The formula for π is referred to as ''Madhava–[[Isaac Newton|Newton]] series'' or ''Madhava–[[Gottfried Leibniz|Leibniz]] series'' or [[Leibniz formula for pi]] or Leibnitz–Gregory–Madhava series.<ref>{{Cite journal|last=Pascal Sebah|author2=Xavier Gourdon|year=2004|title=Collection of series for pi|url=http://math.bu.edu/people/tkohl/teaching/spring2008/piSeries.pdf|access-date=10 February 2010}}</ref> These further names for the various series are reflective of the names of the [[Western world|Western]] discoverers or popularizers of the respective 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 of astronomy and mathematics|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. These are then rendered in current mathematical notations.<ref>{{Cite journal|last=A.K. Bag|year=1975|title=Madhava's sine and cosine series|journal=Indian Journal of History of Science|volume=11|issue=1|pages=54–57|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|access-date=11 February 2010|url-status=dead|archive-url=https://web.archive.org/web/20100214195826/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af4_54.pdf|archive-date=14 February 2010}}</ref><ref>{{Cite book|last=C.K. Raju|title=Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transsmission of the Calculus from India to Europe in the 16 c. CE|publisher=Centre for Studies in Civilistaion|location=New Delhi|year=2007|series=History of Science, Philosophy and Culture in Indian Civilisation|volume=X Part 4|pages=114–120|isbn=978-81-317-0871-2}}</ref>

''Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius.  Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc. ''

Madhava had already computed the value of π using his series formula for π.<ref name="Raju">{{Cite book|last=C.K. Raju|title=Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE|publisher=Centre for Studies in Civilizations|location=Delhi|year=2007|series=History of Philosophy, Science and Culture in Indian Civilization|volume=X Part 4|page=119}}</ref> Using this value of π, namely 3.1415926535922, the radius ''R'' is computed as follows:

:''R'' = 2 &times; 5400 / &pi; = 3437.74677078493925 = 3437 [[arcminute]]s 44 [[arcsecond]]s 48 sixtieths of an [[arcsecond]] = 3437&prime; 44&prime;&prime; 48&prime;&prime;&prime;.

| &nbsp;&nbsp; R &times; (π / 2)³ / 3! &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁵ / 5!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁷ / 7!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁹ / 9!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)¹¹ / 11!  &nbsp;&nbsp;

| &nbsp;&nbsp; R &times; (π / 2)² / 2! &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁴ / 4!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁶ / 6!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)⁸ / 8!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)¹⁰ / 10!  &nbsp;&nbsp;

|  &nbsp;&nbsp; R &times; (π / 2)¹² / 12!  &nbsp;&nbsp;

Madhava's arctangent series is stated in verses 2.206 &ndash; 2.209 in ''Yukti-dipika'' commentary (''Tantrasamgraha-vyakhya'') by [[Sankara Variar]]. A translation of the verses is given below.<ref>{{Cite book|last=C.K. Raju|title=Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transsmission of the Calculus from India to Europe in the 16 c. CE|publisher=Centre for Studies in Civilistaion|location=New Delhi|year=2007|series=History of Science, Philosophy and Culture in Indian Civilisation|volume=X Part 4|page=231|isbn=978-81-317-0871-2}}</ref>

Since  ''c'' =  &pi; ''d'' this can be reformulated as a formula to compute π as follows.

This is obtained by substituting ''q'' = <math>1/\sqrt{3}</math> (therefore ''θ'' = &pi; / 6) in the power series expansion for tan⁻¹ ''q'' above.

{{comparison_pi_infinite_series.svg|400px{{!}}none|two Madhava series (the one with {{radic|12}} in dark blue) and}}

*{{Cite book|last=Kim Plofker|author-link=Kim Plofker|title=Mathematics in India |title-link=Mathematics in India |publisher=Princeton University Press|location=Princeton|year=2009|pages=217–254|isbn=978-0-691-12067-6}}

*"The discovery of the series formula for &pi; by Leibniz, Gregory, and Nilakantha" by Ranjan Roy in : &nbsp;{{Cite book|title=Sherlock Holmes in Babylon and other tales of mathematical history|editor1=Marlow Anderson |editor2=Victor Katz |editor3=Robin Wilson |publisher=[[The Mathematical Association of America]]|year=2004|pages=111–121|isbn=0-88385-546-1}}