EQUAÇÕES GRACELI [10]

P = PROGRESSÃO.

K = VARIÁVEL COMPLEXA.

$\int \limits _{0}^{1}\!{\frac {x^{n-2}\ln \ln {\frac {1}{x}}}{1+x^{2}+x^{4}+\cdots +x^{2n-2}}}\,dx\,=\int \limits _{1}^{\infty }\!{\frac {x^{n-2}\ln \ln {x}}{1+x^{2}+x^{4}+\cdots +x^{2n-2}}}\,dx=$ COS $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}$ $\sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}$ $(n-1)!={\frac {n!}{n}}$ / [PK] $\psi _{n}(z)$ $\sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}$ $\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ [7]

$\int \limits _{0}^{1}\!{\frac {\ln \ln {\frac {1}{x}}}{1+x+x^{2}}}\,dx=\int \limits _{1}^{\infty }\!{\frac {\ln \ln {x}}{1+x+x^{2}}}\,dx={\frac {\pi }{\sqrt {3}}}\ln {\biggl \{}{\frac {\Gamma {(2/3)}}{\Gamma {(1/3)}}}{\sqrt[{3}]{2\pi }}{\biggr \}}$

COS $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}$ $\sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}$ $z!=\Gamma (z+1)=\int _{{0}}^{{\infty }}t^{z}e^{{-t}}\,dt$ [PK] $\psi _{n}(z)$ $[$ $\sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}$ $\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ [7]

$\int \limits _{0}^{1}\!{\frac {\ln \ln {\frac {1}{x}}}{1-x+x^{2}}}\,dx=\int \limits _{1}^{\infty }\!{\frac {\ln \ln {x}}{1-x+x^{2}}}\,dx={\frac {2\pi }{\sqrt {3}}}\ln {\biggl \{}{\frac {\sqrt[{6}]{32\pi ^{5}}}{\Gamma {(1/6)}}}{\biggr \}}$

$COS$ $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}$ $\sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}$ . $(n-1)!={\frac {n!}{n}}$ / [PK] $\psi _{n}(z)$ $[$ $\sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}$ $\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ [7]

$\int \limits _{0}^{1}{\frac {\ln \ln {\frac {1}{x}}}{(1+x)^{2}}}\,dx=\int \limits _{1}^{\infty }\!{\frac {\ln \ln {x}}{(1+x)^{2}}}\,dx={\frac {1}{2}}{\bigl (}\ln \pi -\ln 2-\gamma {\bigr )},$

$COS$ $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{n}k^{m}z^{k}=\left(z{\frac {d}{dz}}\right)^{m}{\frac {1-z^{n+1}}{1-z}}$ $\sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}$ . $z!=\Gamma (z+1)=\int _{{0}}^{{\infty }}t^{z}e^{{-t}}\,dt$ [PK] $\psi _{n}(z)$ $\sum _{k=0}^{\infty }{\frac {E_{2k}z^{2k}}{(2k)!}}=\sec z,|z|<{\frac {\pi }{2}}$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ [7]

P = PROGRESSÃO.

K = VARIÁVEL COMPLEXA.

$\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}$ $\psi _{n}(z)$ $\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ $(n-1)!={\frac {n!}{n}}$ / [PK] $\psi _{n}(z)$ $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\sum _{n=a+1}^{\infty }{\frac {a}{n^{2}-a^{2}}}={\frac {1}{2}}H_{2a}$ [7] $\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{4}+4a^{4}}}={\dfrac {1+a\pi \coth(a\pi )}{8a^{4}}}$

$\int \limits _{0}^{1}\!{\frac {\,\ln \ln {\frac {1}{x}}\,}{1+x^{2}}}\,dx\,=\,\int \limits _{1}^{\infty }\!{\frac {\,\ln \ln {x}\,}{1+x^{2}}}\,dx\,=\,{\frac {\pi }{\,2\,}}\ln \left\{{\frac {\Gamma {(3/4)}}{\Gamma {(1/4)}}}{\sqrt {2\pi \,}}\right\}$ $\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}$ $\psi _{n}(z)$ $\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ $z!=\Gamma (z+1)=\int _{{0}}^{{\infty }}t^{z}e^{{-t}}\,dt$ [PK] $\psi _{n}(z)$ $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n^{4}+4a^{4}}}={\dfrac {1+a\pi \coth(a\pi )}{8a^{4}}}$ $\displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}$

$\int \limits _{0}^{1}\!{\frac {x^{n-2}\ln \ln {\frac {1}{x}}}{1-x^{2}+x^{4}-\cdots +x^{2n-2}}}\,dx\,=\int \limits _{1}^{\infty }\!{\frac {x^{n-2}\ln \ln {x}}{1-x^{2}+x^{4}-\cdots +x^{2n-2}}}\,dx=$ . $(n-1)!={\frac {n!}{n}}$ / [PK] $\psi _{n}(z)$ $\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}$

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ . $z!=\Gamma (z+1)=\int _{{0}}^{{\infty }}t^{z}e^{{-t}}\,dt$ [PK] $\psi _{n}(z)$ $\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}$

P = PROGRESSÃO.

K = VARIÁVEL COMPLEXA.

$\Gamma (t)=\int _{0}^{\infty }x^{{t-1}}e^{{-x}}dx$ $(n-1)!={\frac {n!}{n}}$ $n \choose k$ / [PK] $\sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)$ $\sum _{k=1}^{\infty }{\frac {\mathrm {sen} \,(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi$ $\sum _{k=0}^{\infty }{\frac {\mathrm {sen} \,[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi$ ,[4] $\displaystyle \sum _{n=-\infty }^{\infty }e^{-\pi n^{2}}={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}$

$\Gamma (t)=\int _{0}^{\infty }x^{{t-1}}e^{{-x}}dx$ $n \choose k$ [PK] $\sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)$ $\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\mathrm {sen} \,z$ $\sum _{k=0}^{\infty }{\frac {\mathrm {sen} \,[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi$ ,[4] $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$

$\Gamma (t)=\int _{0}^{\infty }x^{{t-1}}e^{{-x}}dx$ . $(n-1)!={\frac {n!}{n}}$ / [PK] $\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\mathrm {sen} \,z$ $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$

$\Gamma (t)=\int _{0}^{\infty }x^{{t-1}}e^{{-x}}dx$ . $z!=\Gamma (z+1)=\int _{{0}}^{{\infty }}t^{z}e^{{-t}}\,dt$ [PK] $\sum _{k=0}^{\infty }{\frac {\mathrm {sen} \,[(2k+1)\theta ]}{2k+1}}={\frac {\pi }{4}},0<\theta <\pi$ ,[4] $\displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp \left({\frac {2\pi in^{2}q}{p}}\right)={\dfrac {e^{\pi i/4}}{\sqrt {2q}}}\sum _{n=0}^{2q-1}\exp \left(-{\frac {\pi in^{2}p}{2q}}\right)$

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EQUAÇÕES GRACELI [10]

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