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Alan

$$\mathbf{x}(u,v) = \left(10 \cos t, 10 \sin t, t\exp( -.01 t)\right) \qquad $$

$ \frac{\sqrt 14 \pi}{3} $

${x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1$ ,

$ a^2 = 2 + \sqrt{2} $ , $ b = 1 $ , $ c^2 = 2 - \sqrt{2} $

$ abc = \sqrt{2 + \sqrt{2}} \sqrt{2 - \sqrt{2}} = \sqrt{2} $

$ V = \frac{4\pi}{3} {abc} = \frac{4\pi}{3} \sqrt{2} $

$$V = \int_{x=-1}^{x=1} \int_{y=-\sqrt{4(1-x^2)}}^{y=\sqrt{4(1-x^2)}} \int_{z=x+y-3}^{z=0} \ \mathrm{d}z \ \mathrm{d}y \ \mathrm{d}x $$

$v = f_\lambda(z)$ satisfies $zv'' + (\lambda + 1)v' = v$

$$zv^k + (\lambda + k - 1)v^{k - 1} = v^{k - 2} , k=2,3,...$$

$$\frac{v^{k - 1}}{v^k} = \lambda + k + \frac{z}{(\frac{v^k}{v^{k + 1}})}, k= 1,2,...$$

$$ \frac{v}{v'} = \lambda + 1 + \cfrac{z}{\lambda + 2 + \cfrac{z}{\lambda + 3 + \cfrac{z}{\lambda + 4 + \cfrac{z}{\ddots}}}}$$

$$\frac{v'}{v} = \frac{i}{\sqrt z} \frac{K'_ \lambda (2i \sqrt z)}{K_ \lambda (2i \sqrt z)}$$

$$K_\lambda(z) = f_\lambda (-\frac{z^2}{4})$$

$$K_\lambda(z) = 1 + \sum_{n=1}^\infty \frac{(-1)^n}{n! (\lambda+1)\dots(\lambda + n)}(\frac{z}{2})^{2n} , \lambda \ne -1, -2,...$$

$$J_\lambda(z) = \frac{1}{\Gamma(\lambda + 1)} \{(\frac{z}{2})}^{\lambda} K_\lambda(z) $$

$$\frac{1}{5}\int_\{2\arctan{(\frac{1}{\sqrt5}}\sqrt\frac{1 - 11v - v^2}{1 + v - v^2})}^\{2\arctan(\frac{1}{\sqrt5})} \frac{1}{\sqrt{1 -\frac{9}{25}\sin^2\varphi}}d\varphi$$

$$\Psi\left(\frac{m}{k}\right) = -\gamma -\log(2k) -\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +2\sum_{n=1}^{\lceil (k-1)/2\rceil} \cos\left(\frac{2\pi nm}{k} \right) \log\left(\sin\left(\frac{n\pi}{k}\right)\right)$$

$$A = \pmatrix{1&1\\\ 1&2}$$

$$B = \pmatrix{2&1\\\ 1&1}$$

$$\frac{\zeta_{Q_{\sqrt{-7}}}(2)}{\zeta(2)}$$

$$\frac{24}{7\sqrt{7}}\int_\frac{\pi}{3}^\frac{\pi}{2}\ln|\frac{\tan x + \sqrt{7}}{\tan x - \sqrt{7}}|dx$$

$$\frac{1}{3164}\int_0^1 \frac{x^8(1-x)^8(25 + 816x^2)}{1+x^2} \ dx $$

$$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx = \frac{22}{7} - \pi$$

$$\int_{0}^{\infty}\frac{\sqrt{x}\ln{x}}{x^2+ 1}\mathrm{d}x = \frac{\pi^2\sqrt{2}}{4}$$

$$\varphi\left(e^{-\pi} \right) = \frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}$$

$$\varphi\left(e^{-2\pi} \right) = \frac{\sqrt[4]{6\pi+4\sqrt2\pi}}{2\Gamma(\frac{3}{4})}$$

$$\varphi\left(e^{-3\pi}\right) = \frac{\sqrt[4]{27\pi+18\sqrt3\pi}}{3\Gamma(\frac{3}{4})}$$

$$\varphi\left(e^{-4\pi}\right) =\frac{\sqrt[4]{8\pi}+2\sqrt[4]{\pi}} {4\Gamma(\frac{3}{4})}$$

$$\varphi\left(e^{-5\pi} \right) =\frac{\sqrt[4]{225\pi+ 100\sqrt5 \pi}}{5\Gamma(\frac{3}{4})}$$

$$\frac{\sqrt{\pi}}{\Gamma\left(\frac{3}{4}\right)^2}=\frac{1}{{\int_{0}^\frac{\pi}{2}}\sqrt\sin(x)\sqrt\cos(x)\mathrm{d}x}$$

$$\int_{1}^{\infty}\frac{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{x}}}}}}}}{x^2}\mathrm{d}x \sim \frac{1+\sqrt5}{2}$$

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