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Funciones especiales
(Matemática | Cálculo | Integrales | Funciones especiales)

Algunas de estas funciones las he visto definidas en ambos intervalos (0 a x) y (x a inf). En ese caso, los dos varientes se dan.
gamma = la constante de Euler = 0.5772156649...

(x) = Gamma(x) = (integral)(0 to inf)t (x-1) e -tdt (Función gamma)
B(x,y) = (integral)(0 to 1)t (x-1) (1-t) (y-1)dt
(Función beta)
Ei(x) = (integral)(x to inf)e -t/t dt (Integral exponencial) ó una variente no eqivalente:

Ei(x) = + ln(x) + (integral)(0 to x)(e t - 1)/t dt = gamma + ln(x) + (sum)(n=1..inf)x n/(n*n!)
li(x) = (integral)(2 to x)1/ln(t) dt (Integral del logaritmo)
Si(x) = (integral)(x to inf)sen(t)/t dt (Integral del seno) ó una variente no eqivalente:
Si(x) = (integral)(0 to x)sen(t)/t dt = PI/2 - (integral)(x to inf)sen(t)/t dt

Ci(x) = (integral)(x to inf)cos(t)/t dt (Integral del coseno) ó una variente no eqivalente:
Ci(x) = - (integral)(x to inf)cos(t)/t dt = gamma + ln(x) + (integral)(0 to x) (cos(t) - 1) / t dt

Chi(x) = gamma + ln(x) + (integral)(0 to x)(cosh(t)-1)/t dt (Integral del coseno hiperbólico)
Shi(x) = (integral)(0 to x)senh(t)/t dt (Integral del seno hiperbólico)
Erf(x) = 2/PI (1/2)(integral)(0 to x)e (-t^2) dt = 2/sqrtPI (sum)(n=0..inf) (-1) n x (2n+1) / ( n! (2n+1) ) (Función de error)
FresnelC(x) = (integral)(0 to x)cos(PI/2 t 2) dt
FresnelS(x) = (integral)(0 to x)sen(PI/2 t 2) dt
dilog(x) = (integral)(1 to x)ln(t)/(1-t) dt
Psi(x) = (d/dx)ln(Gamma(x))
Psi(n,x) = nth derivada de Psi(x)
W(x) = inverso de x*e x
L sub n (x) = (e x/n!)( x n e -x ) (n) (Polinomial de Laguerre, grado n; (n) significafo nth derivada)
Zeta(s) = (sum)(n=1..inf) 1/n s

Función beta de Dirichlet B(x) = (sum)(n=0..inf) (-1) n / (2n+1) x

Teoremas con hipertexto tienen pruebas, teoremas relacionados, discusiones, y/u otra información.

  
 
  

 
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