ORTOGONAIS GRACELI.

FUNÇÕES ORTOGONAIS COM O SISTEMA PROGRESSIMAL INFINITESIMAL DE GRACELI.

INTEGRAIS, SOMAS E SÉRIES DE GRACELI.

 

séries e integrais de Graceli. 

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

• NÚMERO DE GRACELI =Gn= PI / 1.1 = 2.8559090
• P = PROGRESSÃO.
• 
  
• Aqui, considera-se que  vale 
•  é um polinômio de Bernoulli.
•  é um número de Bernoulli, e aqui, 
•  é um número de Euler.
•  é a função zeta de Riemann.
•  é a função gama.
•  é uma função poligama.
•  é um polilogaritmo .
•  é o coeficiente binomial
•  denota a exponencial

                                                                                            -S / PW                               

           pg

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

T [t] = ao / pk +  [an . cos [nst pk] / L + bn . SEN [nst] / L pk =

SÉRIES DE FURIER COM ELEMENTOS DA MATEMÁTICA DE GRACELI.

$$
{\displaystyle T(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cdot \cos \left({\frac {n\pi t}{L}}\right)+b_{n}\cdot \operatorname {sen} \left({\frac {n\pi t}{L}}\right)\right]}Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle a_{0}={\frac {1}{L}}\int _{c}^{c+2L}f(t)\,dt}$, ${\displaystyle a_{n}={\frac {1}{L}}\int _{c}^{c+2L}f(t)\cos \left({\frac {n\pi t}{L}}\right)\,dt}$ e, ${\displaystyle b_{n}={\frac {1}{L}}\int _{c}^{c+2L}f(t)\,\operatorname {sen} \left({\frac {n\pi t}{L}}\right)\,dt}$

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 . Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

.Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

.Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

.Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

    Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

:

.Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

$$
{\displaystyle T(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cdot \cos \left({\frac {n\pi t}{L}}\right)+b_{n}\cdot \operatorname {sen} \left({\frac {n\pi t}{L}}\right)\right]}Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

]

${\displaystyle f(t)=A_{0}+\sum _{n=1}^{\infty }A_{n}\cdot \cos(w_{n}t-\theta _{n})}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle A_{0}={\frac {a_{0}}{2}}}$, ${\displaystyle A_{n}={\sqrt {a_{n}^{2}+b_{n}^{2}}}}$ para ${\displaystyle n\geq 1}$, ${\displaystyle \theta _{n}=\arctan({\frac {b_{n}}{a_{n}}})}$ e ${\displaystyle w_{n}={\frac {2n\pi }{T}}}$

Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle \cos(a-b)=\cos(a)\cos(b)+\operatorname {sen}(a)\operatorname {sen}(b)\,}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle f(t)=A_{0}+\sum _{n=1}^{\infty }[A_{n}\cos(\theta _{n})\cos(w_{n}t)+A_{n}\sin(\theta _{n})\sin(w_{n}t)]}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle A_{0}={\frac {a_{0}}{2}},a_{n}=A_{n}\cos(\theta _{n})}$ e ${\displaystyle b_{n}=A_{n}\sin(\theta _{n})}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle \operatorname {sen} z={\frac {e^{iz}-e^{-iz}}{2i}}=-i{\frac {e^{iz}-e^{-iz}}{2}}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle \cos z={\frac {e^{iz}+e^{-iz}}{2}}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}{\frac {e^{iw_{n}t}+e^{-iw_{n}t}}{2}}-ib_{n}{\frac {e^{iw_{n}t}-e^{-iw_{n}t}}{2}}\right]}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle w_{n}={\frac {n\pi }{L}}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[{\frac {a_{n}-ib_{n}}{2}}e^{iw_{n}t}+{\frac {a_{n}+ib_{n}}{2}}e^{-iw_{n}t}\right]}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle a_{-n}=a_{n}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle b_{-n}=-b_{n}}$ e ${\displaystyle b_{0}=0}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle cos(-x)=cos(x)}$ e ${\displaystyle sen(-x)=-sen(x)}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle w_{-n}={\frac {(-n)\pi }{L}}=-w_{n}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

: ${\displaystyle c_{n}={\frac {a_{n}-ib_{n}}{2}}}$ , tal que ${\displaystyle c_{0}={\frac {a_{0}-ib_{0}}{2}}={\frac {a_{0}}{2}}}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle c_{0}={\frac {1}{2L}}\int _{-L}^{L}f(t)\,dt}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle {\frac {1}{2L}}\int _{-L}^{L}f(t)\,(cos(w_{n}t)-isen(w_{n}t))dt}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle c_{n}={\frac {1}{2L}}\int _{-L}^{L}f(t)\,e^{-iw_{n}t}dt}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle f(t)=c_{0}+\sum _{n=1}^{\infty }c_{n}\,e^{iw_{n}t}+\sum _{n=1}^{\infty }c_{-n}\,e^{-iw_{n}t}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

${\displaystyle f(t)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{iw_{n}t}}$Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 para ${\displaystyle n\geq 1}$.Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

, ${\displaystyle a_{n}={\frac {2}{L}}\int _{0}^{L}f(t)\,\operatorname {cos} \left({\frac {n\pi t}{L}}\right)\,dt}$ Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  = para ${\displaystyle n\geq 0}$

 Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =

 Gn  [px] = an cos[-1/ ]f[Gn]=+bn sen 1/ Gn [k[pr]ph]  =