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Madhava of Sangamagrama : biography

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Madhava of Sangamagrama ( ; ), was an Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Cochin, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first in the world to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".

 His discoveries opened the doors to what has today come to be known as Mathematical Analysis.  One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. 

Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.

Contributions

If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by James Gregory in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term).

 This implies that the limit nature of the infinite series was quite well understood by him.  Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series, and rational approximations of infinite series. 

However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.

Infinite series

Main article : Madhava series

Among his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhāṣā, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.

In the text, Jyeṣṭhadeva describes the series in the following manner: 

This yields:

r\theta={\frac {r\sin \theta }{\cos \theta
}}-(1/3)\,r\,{\frac { \left(\sin \theta   \right) ^ 

{3}}{ \left(\cos \theta \right) ^{3}}}+(1/5)\,r\,{\frac {

\left(\sin \theta  \right) ^{5}}{ \left(\cos 

\theta \right) ^{5}}}-(1/7)\,r\,{\frac { \left(\sin \theta

\right) ^{7}}{ \left(\cos \theta  \right) ^{ 

7}}} + \cdots or equivalently:

\theta = \tan \theta - \frac{\tan^3 \theta}{3} + \frac{\tan^5 \theta}{5} - \frac{\tan^7 \theta}{7} + \cdots

This series was traditionally known as the Gregory series (after James Gregory, who discovered it three centuries after Madhava). Even if we consider this particular series as the work of Jyeṣṭhadeva, it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava. Today, it is referred to as the Madhava-Gregory-Leibniz series.

Trigonometry

Madhava also gave a most accurate table of sines, defined in terms of the values of the half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is believed that he may have found these highly accurate tables based on these series expansions:

sin q = q – q3/3! + q5/5! – ...
cos q = 1 – q2/2! + q4/4! – ...

The value of π (pi)

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