How did Madhava come up with the Arctan series?

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Discussion Overview

The discussion centers on the historical context and methods behind Madhava's discovery of the power series for the arctangent function. Participants explore various hypotheses regarding Madhava's mathematical understanding and the influences on his work, including cultural and educational factors.

Discussion Character

  • Exploratory, Historical, Debate/contested

Main Points Raised

  • One participant suggests that Madhava may have discovered the series through a process akin to the fundamental theorem of calculus, questioning whether he had the concepts of uniform convergence or polynomial integration.
  • Another participant humorously proposes that the series was revealed by Vishnu, indicating a cultural or religious interpretation of Madhava's insights.
  • A different viewpoint highlights the lack of detailed explanations in historical Indian mathematics, suggesting that local schools may have had their own methods that were not documented.
  • One participant speculates that Madhava's understanding of infinite geometric series could have influenced his work, particularly in relation to trigonometric functions.
  • References to Wikipedia articles mention that Madhava's methods are discussed in a book by his followers, although details are not provided in the thread.

Areas of Agreement / Disagreement

Participants express differing views on how Madhava might have arrived at the arctangent series, with no consensus on the specific methods or influences involved. The discussion remains open-ended with multiple competing interpretations.

Contextual Notes

Participants note the absence of detailed historical documentation, which limits the understanding of Madhava's methods and the educational context of his time.

imurme8
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Does anyone know how Madhava discovered the power series for the arctangent? I think the standard way is to note that 1-x^2+x^4-\dotsb converges uniformly on (-1,1) to \frac{d}{dt}\tan^{-1}x, and thus applying the fundamental theorem of calculus we may integrate term-by-term. But how did Madhava do it? I don't know that he had the FTOC or a concept of uniform convergence, or even that he knew how to integrate a polynomial.
 
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obviously it was revealed by vishnu.
 
mathwonk said:
obviously it was revealed by vishnu.

That made me laugh.
 
The problem with much of Indian mathematics is that it is in the form: "See!", with no attendant explanations.
Presumably, such explanations were developed in the local schools and research centres, but we have not, unfortunately, been handed down lecture notes and such from those days.

If I were to make a guess, I would think they were well aware of the sum of an infinite geometric series, and that the result given has a close relationship to their understanding of how 1/(1+x^2) appeared within trigonometry.
 

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