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Madhava Series
In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent function (mathematics), 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. Using modern notation, these series are: :\begin \sin \theta &= \theta - \frac + \frac - \frac + \cdots &&= \sum_^\infty \frac\theta^, \\[10mu] \cos \theta &= 1 - \frac + \frac - \frac + \cdots &&= \sum_^\infty \frac\theta^, \\[10mu] \arctan x &= x - \frac + \frac - \frac + \cdots &&= \sum_^\infty \fracx^ \quad \text , x, \leq 1. \end 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 (mathematician), James Gregory in 1671 and Gottfried Wilhelm Leibniz, Gottfried ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Formula For π
In mathematics, the Leibniz formula for , named after Gottfried Wilhelm Leibniz, states that \frac = 1-\frac+\frac-\frac+\frac-\cdots = \sum_^ \frac, an alternating series. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was later independently rediscovered by James Gregory in 1671 and Leibniz in 1673. The Taylor series for the inverse tangent function, often called '' Gregory's series'', is \arctan x = x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac. The Leibniz formula is the special case \arctan 1 = \tfrac14\pi. It also is the Dirichlet -series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and therefore the value of the Dirichlet beta function. Proofs Proof 1 \begin \frac &= \arctan(1) \\ &= \int_0^1 \frac 1 \, dx \\ pt& = \int_0^1\left(\sum_^n (-1)^k x^+\frac\right) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Katapayadi System
''Kaṭapayādi'' system (Devanagari: कटपयादि, also known as ''Paralppēru'', Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered. History The oldest available evidence of the use of ''Kaṭapayādi'' (Sanskrit: कटपयादि) system is from ''Grahacāraṇibandhana'' by Haridatta in 683 CE.Sreeramamula Rajeswara Sarma, THE ''KATAPAYADI'' SYSTEM OF NUMERICAL NOTATION AND ITS SPREAD OUTSIDE KERALA, ''Rev. d'Histoire de Mathmatique'' 18 (2012/ref> It has been used in ''Laghu·bhāskarīya·vivaraṇa'' written by '' Sankara Narayana, Śaṅkara·nārāyaṇa'' in 869 CE. In some astronomical texts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arcsecond
A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of a degree. Since one degree is of a turn, or complete rotation, one arcminute is of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth's circumference is very near . A minute of arc is of a radian. A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol , is a unit of angular measurement equal to of a minute of arc, of a degree, of a turn, and (about ) of a radian. These units originated in Babylonian astronomy as sexagesimal (base 60) subdivisions of the degree; they are used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, navigation, land surveying, and marksmanship. To express even smaller angles, standard SI prefixes can be employed; the milliarcse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jyā, Koti-jyā And Utkrama-jyā
Jyā, koṭi-jyā and utkrama-jyā are three trigonometric functions introduced by Indian mathematics, Indian mathematicians and astronomers. The earliest known Indian treatise containing references to these functions is Surya Siddhanta. These are functions of arcs of circles and not functions of angles. Jyā and koti-jyā are closely related to the modern trigonometric functions of sine and cosine. In fact, the origins of the modern terms of "sine" and "cosine" have been traced back to the Sanskrit words jyā and koti-jyā. Definition Let 'arc AB' denote an Arc (geometry), arc whose two extremities are A and B of a circle with center 'O'. If a perpendicular BM is dropped from B to OA, then: * ''jyā'' of arc AB = BM * ''koti-jyā'' of arc AB = OM * ''utkrama-jyā'' of arc AB = MA If the radius of the circle is ''R'' and the length of arc AB is ''s'', the angle subtended by arc AB at O measured in radians is θ = ''s'' / ''R''. The three Indian functions are related to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sankara Variar
Sankara Variyar (; .) was an astronomer-mathematician of the Kerala school of astronomy and mathematics. His family were employed as temple-assistants in the temple at near modern Ottapalam. Mathematical lineage He was taught mainly by Nilakantha Somayaji (1444–1544), the author of the Tantrasamgraha and Jyesthadeva (1500–1575), the author of Yuktibhāṣā. Other teachers of Shankara include Netranarayana, the patron of Nilakantha Somayaji and Chitrabhanu, the author of an astronomical treaties dated to 1530 and a small work with solutions and proofs for algebraic equations. Works The known works of Shankara Variyar are the following: * ''Yukti-dipika'' - an extensive commentary in verse on ''Tantrasamgraha'' based on ''Yuktibhāṣā''. * ''Laghu-vivrti'' - a short commentary in prose on ''Tantrasamgraha''. * '' Kriya-kramakari'' - a lengthy prose commentary on Lilavati of Bhaskara II. * An astronomical commentary dated 1529 CE. * An astronomical handbook completed ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tantrasamgraha
Tantrasamgraha, or Tantrasangraha, (literally, ''A Compilation of the System'') is an important astronomy, astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit divided into eight chapters. Tantrasamgraha had spawned a few commentaries: ''Tantrasamgraha-vyakhya'' of anonymous authorship and ''Yuktibhāṣā'' authored by Jyeshtadeva in about 1550 CE. Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Madhava of Sangamagrama, Sangamagrama Madhava. In his ''Tantrasangraha'', Nilakantha revised Aryabhata's model for the planets Mercury (planet), Mercury and Venus. According to George Gheverghese Joseph, George G Joseph his equation of the Ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
