Thread:The Zog./@comment-34296765-20151031065114/@comment-34296765-20151106032316

$$\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}} where \{\dots,-2\pi i, 0, 2\pi i, 4\pi i,\dots\} = \{2\pi ki| k\in\mathbb Z\} and$$ $$ \pi=\textstyle \cfrac{4}{1+\textstyle \frac{1^3}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^3}{2+\textstyle \frac{9^2}{2+\ddots}}}}}} =3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}} =\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}}$$ $$whereas \frac{1}{\pi} = \frac{12}{640320^{3/2}} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}} + \pi $$ $$= \sum_{k=5}^\infty \frac{1}{29^k} \left( \frac{7}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right) \frac{2\varphi}{\log 2} \approx 4.669 \frac{2\varphi+1}{\log 2+1}$$ \approx 2.502 \frac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v}\cdot\nabla ) \mathbf{v} = -\nabla p + \nu\Delta \mathbf{v} +\mathbf{f}(\boldsymbol{x},t)