Numero di Eulero – ValCon.It

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$\displaystyle e=\lim_{n \rightarrow +\infty}\left(1+\frac{1}{n}\right)^n$

$\displaystyle e^x=\lim_{n \rightarrow +\infty}\left(1+\frac{x}{n}\right)^n$

$\displaystyle e=\sum_{n=0}^{+\infty}\frac{1}{n!}$

$e^{i x}=\cos x +i \sin x$

$e^{i \pi}+1=0$

$\displaystyle \frac{\partial e^x}{\partial x}= e^x$

$\displaystyle e^x=\sum_{n=0}^{+\infty}\frac{x^n}{n!}$

$\displaystyle n!\approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^{n}$

$\displaystyle f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2 {\sigma}^2}}$

$\displaystyle\int e^x\ dx = e^x + c$

$\displaystyle\int_{0}^{+\infty}\,e^{-x^2}\,dx=\sqrt{\frac{\pi}{4}}$

$\displaystyle\int_{-\infty}^{+\infty}\,e^{-x^2}\,dx=\sqrt{\pi}$

$\displaystyle \int_{-\infty}^{+\infty}\,e^{-\frac{x^2}{2}}\,dx=\sqrt{2\pi}$