#madhava — Public Fediverse posts

Two physicists, Arnab Priya Saha and Aninda Sinha from the Indian Institute of Science (IISc) Bengaluru, inadvertently discovered a new formula for calculating \(\pi\) while working on string theory. Their findings were published in Physical Review Letters in January 2024.

This formula generates an infinitely long sum. What's remarkable is that it depends on a factor \(\lambda\), a freely adjustable parameter.

Since there are infinitely many possible values for \(\lambda\), Saha and Sinha have effectively discovered infinite formulas for \(\pi\). Interestingly, when \(\lambda\) approaches infinity, the equation corresponds to Madhava's formula, discovered more than 600 years ago.
\[\pi = 4 + \sum_{n=1}^\infty {1\over n!} \biggl({1\over n+\lambda} - {4\over 2n+1}\biggr)\biggl({(2n+1)^2 \over 4(n+\lambda)} - n \biggr)_{n-1}\]

where \(\lambda\) is an arbitrary complex number and the Pochhammer symbol \((x)_n := x(x+1)\cdots(x+n-1)\).

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