Madhava series

In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent 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.

However, in the writing of later Kerala school mathematicians Nilakantha Somayaji (1444 – 1544) and Jyeshthadeva (c. 1500 – c. 1575) one can find unambiguous attributions of these series to Madhava.

These later works also include proofs and commentary which suggest how Madhava may have arrived at the series.

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.

[8][9] Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar.

Substituting these in the last expression and simplifying we get which is the infinite power series expansion of the sine function.

Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.437 in Yukti-dipika): vi-dvān, tu-nna-ba-la, ka-vī-śa-ni-ca-ya, sa-rvā-rtha-śī-la-sthi-ro, ni-rvi-ddhā-nga-na-rē-ndra-rung .

Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar.

Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times).

Substituting these in the last expression and simplifying we get which gives the infinite power series expansion of the cosine function.

The last line in the verse ′as collected together in the verse beginning with stena, stri, etc.′ is a reference to a reformulation introduced by Madhava himself to make the series convenient for easy computations for specified values of the arc and the radius.

Let R be the radius of a circle one quarter of which measures C. Then, as in the case of the sine series, Madhava gets

Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.438 in Yukti-dipika): The six stena, strīpiśuna, sugandhinaganud, bhadrāngabhavyāsana, mīnāngonarasimha, unadhanakrtbhureva.

Madhava's arctangent series is stated in verses 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar.

That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc.

Substituting these in the last expression and simplifying we get Letting tan θ = q we finally have The second part of the quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.