The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Malabar, Kerala, South India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati,Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, calledTantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1500 – c. 1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala. Contributions
Infinite Series and Calculus
The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series: for | x | < 1
This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (965–1039). The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result: for large n. This result was also known to Alhazen.
They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for sinx, cosx, andarctanx .The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where
where, for r = 1, the series reduce to the standard power series for these trigonometric functions, for example: and
(The Kerala school themselves did not use the "factorial" symbolism.) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle), was not yet developed.) They also made use of the series expansion of arctanx to obtain an infinite series expression (later known as Gregory series) for π:
Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, fi(n + 1), (for nodd, and i = 1, 2, 3) for the series: