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Función gamma
(Matemática | Cálculo | Integrales | Funciones especiales | Función gamma)
Curva de Gamma


Gamma(x) = (integral)(0 to inf) e -t t(x-1) dt

Gamma(x) = rx(integral)(0 to inf) e -rt t(x-1) dt

Gamma(x) = 2(integral)(0 to inf) e(-t^2) t(2x-1) dt

( Gamma(x)Gamma(y) ) / ( Gamma(x) + Gamma(y) ) = Beta(x,y)

Gamma(x+1) = x Gamma(x)

Gamma(x+1) = x!

Gamma(0+) = +inf

Gamma(1/2) = sqrt(PI)

Gamma(z) Gamma(1-z) = PI csc(PI z) Gamma'(1) = - gamma (el constante de Euler = 0.577215...)

Gamma'(x) = Gamma(x) lim (n-->inf) [ ln(n) - sum (k=0..n-1) 1/(x+k) ]

Expansiones de la función Gamma
Gamma(x+1) = lim (k-->inf) kx 1*2*3*..*k
(x+1)(x+2)*..*(x+k)
La serie asintótica del Stirling
Gamma(x+1) = sqrt(2PIx) xx e -x { 1 + 1/(12x) + 1/(288 x2) - 139/(51840 x3) - 571/(2488320 x4) + 163879/(209018880 x5) + 5246819/(75246796800 x6) - 534703531/(902961561600 x7) - ...} (este solamente es una aproximación. Ver nota (en inglés).)

Funciones de gamma incompletas:
Gamma(x, a) = (integral)(a to inf) e -t t(x-1) dt
gamma(x, a) = (integral)(0 to a) e -t t(x-1) dt

Gamma(x, a) + gamma(x, a) = Gamma(x)

  
 
  

 
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