User:GX, May 1971/Math/Analysis

= $Factorials$ =

$Gaussians$

 * $$\mathcal{G}(n)\ =\ \int_0^\infty{e^{-x^n}\ dx}$$

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 * $$\mathcal{G}(0)\ =\ \infty\ \qquad\qquad\qquad\quad\ \mathcal{G}(\infty)\ =\ 1\ \qquad\qquad\qquad\quad\ \mathcal{G}'\left(2\tfrac16\right)\ =\ 0^{^+}$$

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$Factorials$



 * $$n!\ =\ \mathcal{G}\Big(\tfrac1n\Big)\ =\ \int_0^\infty{e^{-\sqrt[n]x}\ dx} \qquad\qquad\qquad\qquad\qquad\qquad \Gamma^'\Big(1\tfrac6{13}\Big)\ =\ 0^{^-}$$

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 * $$n!\ =\ \int_0^\infty{e^{-\sqrt[n]x}\ dx}\ =\ \int_0^\infty{\frac{\ x^n}{e^x}}\ dx\ =\ \int_0^1{\Big|\ln x\Big|^n\ dx}\ =\ \int_0^1{\Big(\!\!-\ln x\Big)^n\ dx}$$


 * $$n!\ =\ \int_0^1{\ln^n \Big(\tfrac1x\Big)\ dx}\ =\ \int_1^\infty{\frac{\ln^n x}{x^2}\ dx}\ \simeq\ \sum_{k = 1}^\infty{\frac{\ln^n k}{k^2}}\ =\ \bigg|\zeta^{(n)}(2)\bigg|$$

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= $Continued Fractions. Nested Radicals$ =

$The factorial of a positive number is the Gaussian of its reciprocal or multiplicative inverse:$

 * $$\underset{_{\text{k}\,=\,0}}\overset{n}\mathbf\Xi\ \Bigg(a_{_\text{k}}\ ,\ b_{_\text{k}},\ \frac1{N_{_\text{k}}}\Bigg)\ =\ \sqrt[^{N_{_\text{0}}}]{a_{_\text{0}}\ +\ b_{_\text{0}}\sqrt[^{N_{_\text{1}}}]{a_{_\text{1}}\ +\ b_{_\text{1}}\sqrt[^{N_{_\text{2}}}]{\ldots\ \sqrt[^{N_{_{n}}}]{a_{_{n}}}}}}$$

$Nested Radicals$

 * $$\underset{_{\text{k}\,=\,0}}\overset{n}\mathbf\Xi\ \Big(a_{_\text{k}}\ ,\ b_{_\text{k}},\ -1\Big)\ =\ \cfrac1{a_{_0}\ +\ \cfrac{b_{_0}}{a_{_1}\ +\ \cfrac{b_{_1}}{\ddots\ {a_{_n}}}}}$$

$Continued Fractions$

 * $$\underset{_{\text{k}\,=\,0}}\overset{n}\mathbf\Xi\ \Big(a_{_\text{k}}\ ,\ 1,\ 1\Big)\ =\ \sum_{k\,=\,0}^n a_{_\text{k}} \qquad\qquad;\qquad\qquad \underset{_{\text{k}\,=\,0}}\overset{n}\mathbf\Xi\ \Big(0,\ a_{_\text{k}}\ ,\ 1\Big)\ =\ \prod_{k\,=\,0}^n a_{_\text{k}}$$

$Sums & Products$

 * $$\underset{_{\text{k}\,=\,0}}\overset{n}\mathbf\Xi\ \Big(a_{_\text{k}}\ ,\ x,\ 1\Big)\ =\ \sum_{k\,=\,0}^n a_{_\text{k}}\ x^k\ =\ P_n(x)$$

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= $Exponentiation and Tetration$ =