EQUAÇÕES GRACELI [15]

$\int _{-\infty }^{+\infty }\mathrm {exp} (-{\frac {1}{2}}t^{2})\mathrm {d} t={\sqrt {2\,\pi }}$ . [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$\Phi (x)={\frac {1}{\sqrt {2\;\pi }}}\int _{-\infty }^{x}\mathrm {e} ^{-{\frac {1}{2}}t^{2}}\mathrm {d} t$ [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$\Phi (x)={\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)$ [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$\operatorname {erf} (x)=2\Phi \left(x{\sqrt {2}}\right)-1$ [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$\phi (t)=\int _{-\infty }^{+\infty }{\rm {e}}^{{\rm {i}}tx}\mathrm {d} \Phi (x)=\int _{-\infty }^{+\infty }{\rm {e}}^{{\rm {i}}tx-{\frac {x^{2}}{2}}}\mathrm {d} x={\rm {e}}^{-{\frac {t^{2}}{2}}}$ [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$M(t)=\mathbb {E} [e^{tX}]={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\rm {e}}^{xt}{\rm {e}}^{-{\frac {x^{2}}{2}}}\mathrm {d} x={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\rm {e}}^{-({\frac {(x-t)^{2}-t^{2}}{2}})}\mathrm {d} x={\rm {e}}^{\frac {t^{2}}{2}}\,{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{\rm {e}}^{-{\frac {x^{2}}{2}}}\mathrm {d} x={\rm {e}}^{\frac {t^{2}}{2}}$ .^[ [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$f(t)={\frac {1}{\sigma {\sqrt {2\;\pi }}}}\;\;\mathrm {e} ^{-{\frac {1}{2}}{\frac {(t-\mu )^{2}}{\sigma ^{2}}}}$ , [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

$F(x)={\frac {1}{\sigma {\sqrt {2\;\pi }}}}\int _{-\infty }^{x}\mathrm {e} ^{-{\frac {1}{2}}{\frac {(t-\mu )^{2}}{\sigma ^{2}}}}\mathrm {d} t$ , [ ${\frac {\pi }{2}}t^{2}$ ] dt $[pk] n!, ] [$ $\ln(n!)=\sum _{{k=1}}^{n}{\ln(k)}$ $\zeta (s)$ / [ $\pi$ / pk]

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EQUAÇÕES GRACELI [15]

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