FUNÇÕES ZETA COM FATORIAL E PROGRESSÕES.

FUNÇÃO ZETA COM FATORIAL E PROGRESSÕES.

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta (s)=\sum _{k=1}^{\infty }f(k)^{s}$ / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta _{K}(s)=\sum _{I\subseteq O_{K}}(N_{K/\mathbb {Q} }(I))^{-s}$ / $n!$ / [PK] $\zeta (s)$ [n]

$Z_{V,Q}(s)$ / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta _{V,p}\left(p^{-s}\right).$ / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.$ / $n!$ / [PK] $\zeta (s)$ [n]

FUNÇÃO ZETA COM FATORIAL E PROGRESSÕES.

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta (s)=\sum _{k=1}^{\infty }f(k)^{s}$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta _{K}(s)=\sum _{I\subseteq O_{K}}(N_{K/\mathbb {Q} }(I))^{-s}$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

$Z_{V,Q}(s)$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta _{V,p}\left(p^{-s}\right).$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

$\zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.$ $[$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

FUNÇÃO ZETA COM FATORIAL E PROGRESSÕES.

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}.$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

$\zeta (s)=\sum _{k=1}^{\infty }f(k)^{s}$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

$\zeta _{K}(s)=\sum _{I\subseteq O_{K}}(N_{K/\mathbb {Q} }(I))^{-s}$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

$Z_{V,Q}(s)$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

$\zeta _{V,p}\left(p^{-s}\right).$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

$\zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.$ $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

EQUAÇÃO GRACELI COM FATORIAL E ZETA.

$n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ -COS] $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

[-COS] A $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + $n!$ / [PK] $\zeta (s)$ /2 =

EQUAÇÃO GRACELI COM FATORIAL E ZETA.

COS B $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n]

+ SEN B [/] $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

COS B A $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $B An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + $n!$ / [PK] $\zeta (s)$ /2 + SEN B / A $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $B An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + $n!$ / [PK] $\zeta (s)$ /2

EQUAÇÃO GRACELI COM FATORIAL E ZETA.

$n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

COS $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + SEN $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

A $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + $n!$ / [PK] $\zeta (s)$ /2 =

EQUAÇÃO GRACELI COM FATORIAL E ZETA.

$n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] =

A $n!$ / [PK] $\zeta (s)$ $d^{k}y \over dx^{k}$ $n \choose k$ $B An$ $n!$ / [PK] $\zeta (s)$ [-pk] / $n!$ / [PK] $\zeta (s)$ [n] + $n!$ / [PK] $\zeta (s)$ /2 =

FUNÇÃO GRACELI.

P = PROGRESSÃO.

K = VARIÁVEL COMPLEXA.

$\zeta (s)$ = [-1[ $n!$ ]] / f [-1[ $n!$ ]] / [u [ $n!$ / PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ ] / û [ $n!$ / PK $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ / [ $n!$ ]- x dx

$\zeta (s)$ = f [u [ $n!$ / PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ ] / û [ $n!$ / PK $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ /[ $n!$ ]- x dx

$\zeta (s)$ = 1 / f [u [ $n!$ / PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ ] / û [ $n!$ / PK $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ - x dx [-1]=

$\zeta (s)$ = f [u [ $n!$ / PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ ] / û [ $n!$ / PK $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ - x [+1] dx = 0

\zeta (s)

f [u [

n!

/ PK]

\zeta (s)

\mathrm {e^{i2\pi \phi }}

] / û [

n!

/ PK

\zeta (s)

\mathrm {e^{i2\pi \phi }}

- x dx

[ $n!$ / PK ] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ = COS [ $n!$ / PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ [+,-] SIN [ $n!$ /PK] $\zeta (s)$ $\mathrm {e^{i2\pi \phi }}$ =

[ $n!$ / PK] $\Gamma (z)$ $\mathrm {e^{i2\pi \phi }}$ = COS [ $n!$ /PK] $\Gamma (z)$ $\mathrm {e^{i2\pi \phi }}$ [+,-] SIN [ $n!$ / PK] $\Gamma (z)$ $\mathrm {e^{i2\pi \phi }}$ =

[ $n!$ /PK] $\psi _{n}(z)$ $\mathrm {e^{i2\pi \phi }}$ = COS [ $n!$ /PK] $\psi _{n}(z)$ $\mathrm {e^{i2\pi \phi }}$ [+,-] SIN [ $n!$ / PK] $\psi _{n}(z)$ $\mathrm {e^{i2\pi \phi }}$ =

[ $n!$ / PK $\operatorname {Li} _{s}(z)$ $\mathrm {e^{i2\pi \phi }}$ = COS [ $n!$ / PK] $\operatorname {Li} _{s}(z)$ $\mathrm {e^{i2\pi \phi }}$ [+,-] SIN [ $n!$ / PK $\operatorname {Li} _{s}(z)$ $\mathrm {e^{i2\pi \phi }}$ =

[ $n!$ /PK] $E_{n}$ $\mathrm {e^{i2\pi \phi }}$ = COS [ $n!$ /PK] $E_{n}$ $\mathrm {e^{i2\pi \phi }}$ [+,-] SIN [ $n!$ /PK] $E_{n}$ $\mathrm {e^{i2\pi \phi }}$ =

Esta lista de séries matemáticas contém fórmulas para somas finitas e infinitas. Ela pode ser usada em conjunto com outras ferramentas para avaliar somas.

Aqui, considera-se que $0^{0}$ vale $1$
$B_{n}(x)$ é um polinômio de Bernoulli.
$B_{n}$ é um número de Bernoulli, e aqui, $B_{1}=-{\frac {1}{2}}.$
$E_{n}$ é um número de Euler.
$\zeta (s)$ é a função zeta de Riemann.
$\Gamma (z)$ é a função gama.
$\psi _{n}(z)$ é uma função poligama.
$\operatorname {Li} _{s}(z)$ é um polilogaritmo .
$n \choose k$ é o coeficiente binomial
$\exp(x)$ denota a exponencial de $x$

n!

= fatorial

G { f [x] } = û [

n!

/ PK ]

\zeta (s)

\mathrm {e^{i2\pi \phi }}

[x] = [

n!

/ PK]

\zeta (s)

\mathrm {e^{i2\pi \phi }}

f [u [

n!

/ PK]

\zeta (s)

\mathrm {e^{i2\pi \phi }}

] / û [

n!

/ PK

\zeta (s)

\mathrm {e^{i2\pi \phi }}

- x dx

integria com fatoriais e progressões.

Integrais Indefinidas de Funções Simples

Funções Racionais

Ver artigo principal: Lista de integrais de funções racionais

$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ para }}n\neq -1$ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{x}}\,dx=\ln {\left|x\right|}+C$ [10][ -COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}{\mbox{arc tg }}{\frac {x}{a}}+C$ [ -COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{a^{2}-x^{2}}}dx={\frac {1}{2a}}\ln \left|{\frac {a+x}{a-x}}\right|+C$ [11] [-COS+ $n!$ / PK] $\zeta (s)$ ]

Logaritmos

Ver artigo principal: Lista de integrais de funções logarítmicas

$\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \ln x\,dx=x(\ln x-1)+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]

Funções Exponenciais

Ver artigo principal: Lista de integrais de funções exponenciais

$\int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int e^{x}\,dx=e^{x}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]

Funções Irracionais

Ver artigo principal: Lista de integrais de funções irracionais

$\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx={\mbox{arc sen }}{\frac {x}{a}}+C$ [+ $n!$ / PK] $\zeta (s)$ ]
$\int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx=\arccos {\frac {x}{a}}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{x{\sqrt {x^{2}-a^{2}}}}}dx={\frac {1}{a}}{\mbox{arc sec }}\left|{\frac {x}{a}}\right|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{\sqrt {a^{2}+x^{2}}}}dx=\ln |x+{\sqrt {x^{2}+a^{2}}}|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{\sqrt {x^{2}-a^{2}}}}dx=\ln |x+{\sqrt {x^{2}-a^{2}}}|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|+C$
$\int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]

Funções Trigonométricas

Ver artigo principal: Lista de integrais de funções trigonométricas

$\int \cos {x}\,dx=\operatorname {sen} {x}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{sen }}{x}\,dx=-\cos {x}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{tg }}{x}\,dx=-\ln {\left|\cos {x}\right|}+C$ [12] [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cossec }}{x}\,dx=\ln {\left|{\mbox{cossec }}{x}-{\mbox{cotg }}{x}\right|}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \sec {x}\,dx=\ln {\left|\sec {x}+{\mbox{tg }}{x}\right|}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cotg }}{x}\,dx=\ln {\left|{\mbox{sen }}{x}\right|}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \sec {x}{\mbox{tg }}{x}\,dx=\sec {x}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cossec }}{x}{\mbox{cotg }}{x}\,dx=-{\mbox{cossec }}{x}+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \sec ^{2}x\,dx={\mbox{tg }}x+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cossec}}^{2}x\,dx=-{\mbox{cotg }}x+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{sen}}^{2}x\,dx={\frac {1}{2}}(x-{\mbox{sen }}x\cos x)+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \cos ^{2}x\,dx={\frac {1}{2}}(x+{\mbox{sen }}x\cos x)+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]

Funções Hiperbólicas

Ver artigo principal: Lista de integrais de funções hiperbólicas

$\int {\mbox{senh }}x\,dx=\cosh x+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int \cosh x\,dx={\mbox{senh }}x+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{tgh }}x\,dx=\ln(\cosh x)+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cossech}}\,x\,dx=\ln \left|{\mbox{tgh }}{x \over 2}\right|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{sech}}\,x\,dx={\mbox{arctg }}({\mbox{senh }}x)+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]
$\int {\mbox{cotgh }}x\,dx=\ln |{\mbox{senh }}x|+C$ [-COS+ $n!$ / PK] $\zeta (s)$ ]