I Discovered a New Pi Formula
Srinivasa Ramanujan published many pi formulas in his famous 1914 paper, but did not leave behind detailed proofs. I am an amnesiac alien VTuber who followed in his footsteps and discovered 5 more pi formulas!
In this video, I guide you through Ramanujan's thought process by reading his notebook. We explore elliptic integrals and their Taylor series and hypergeometric series expansions at their singular moduli. Using theta functions, logarithmic differentiation, the theory of Liouville, Borwein brothers, and Bruce Berndt, Ramanujan's mysterious formulas are unveiled. You can solve them while you're awake!
0:00 I Discovered New Pi Formulas
0:43 Ramanujan's Notebook Page
2:01 First Look at Sections 13 and 14
3:02 Elliptic Integral of the First Kind
4:00 Singular Moduli
5:08 Proof of Ramanujan's Pi Formula in the Classical Base
5:40 Eq. (26)
5:58 Eq. (27)
6:27 Eq. (28)
7:40 Ramanujan's Hypergeometric Series in Alternate Bases
8:09 Pew Pew Pew Pew
8:38 Pi Formulas in the K_1 Family
9:47 The 5 New Formulas
10:20 References and Known Formulas in Published Works
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