Formulas: Difference between revisions
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The most beautiful formulas: | The most beautiful formulas: | ||
Ramanujan's inverse pi formula: | |||
<math> | |||
\pi^{-1} = \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{99^{4k}} | |||
</math> | |||
Gaussian integral: | |||
<math> | <math> | ||
Revision as of 08:13, 4 March 2019
The most beautiful formulas:
Ramanujan's inverse pi formula:
$ \pi^{-1} = \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{99^{4k}} $
Gaussian integral:
$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{2 \pi} $
Ramanujan sum:
$ \sum_{k=1}^{\infty} k = - \frac{1}{12} $
Zeta-regularized product:
$ \prod_{k=1}^{\infty} k = \sqrt{2 \pi} $
Euler formula:
$ e^{i \pi} + 1 = 0 $
Archimedes' Recurrence Formula:
$ a_{2n} = \frac{2 a_n b_n}{a_n + b_n} $
$ b_{2n} = \sqrt{a_{2n} b_n} $