formula rio de funciones especiales para calculo iv
TRANSCRIPT
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7/31/2019 Formula Rio de Funciones Especiales Para Calculo IV
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FORMULARIO DE FUNCIONES ESPECIALES
1. ( x )n = x ( x + 1 ) ( x + n 1 ) , x R 2. ( x )n = x ( x + 1 )n1 3. ( x )n = ( x )n1 ( x + n 1 )
4. ( x ) 0 = 1 , si x = 0 5.
1
2
n
=( 2n )!
2 2n n!6. ( 1 )n = n!
7. ( x ) =
0
e t tx1 dt , x > 0 8. ( 1 ) = 1 9.
1
2
=
10.
n +
1
2
= 1
2n
12
11. ( x + 1 ) = x ( x ) 12. ( x + n ) = ( x )n ( x )
13. ( x ) ( 1 x ) = sen( x )
, x / Z 14. ( x ) = 20
et2
t2x1dt , x > 0 15. ( n + 1 ) = n! , n N
16. B ( x, y ) =
10
t x1 ( 1 t )y1 dt x, y > 0 17. B ( x, y ) = B ( y, x ) 18. B ( x, y ) = ( x ) ( y ) ( x + y )
19. B ( x, y ) = 2
/20
cos2x1 ( )sen2y1 ( ) d 20.
n
k
=
n!
k! ( n k )! con n > k 21. ( a + b )n
=
nk=0
n
k
a kb n
22. Ecuacion de Legendre
1 x2
y 2xy + ( + 1 ) y = 0 , R+
23. f1 ( x ) = 1 +
n=1
(1 )n [ ( 2 ) ( 2n + 2 ) ] [ ( + 1 ) ( + 3 ) ( + 2n 1 ) ]( 2n )!
x 2n
24. f2 ( x ) = x +
n=1
(1 )n [ ( 1 ) ( 3 ) ( 2n + 1 ) ] [ ( + 2 ) ( + 4 ) ( + 2n ) ]( 2n + 1)!
x 2n+1
25. Formula General : Pn ( x ) =
[|n/2 |]k=0
(1 ) k ( 2n 2k )!2 nk! ( n k )! ( n 2k )! x
n2k , n = 0, 1, 2, 3, . . .
26. Formula de Rodriguez : Pn ( x ) =1
2 nn!D nx
x 2 1 n 27.
11
Pn ( x ) Pm ( x ) dx = 0 m = n
28. Formula Generatriz :
1 2xt + t 2 1/2 = n=0
Pn ( x ) tn 29.
1
1
[ Pn ( x ) ]2
dx =2
2n + 1
30. ( n + 1 ) Pn+1 ( x ) ( 2n + 1 ) xPn ( x ) + nPn1 ( x ) = 0 31. P n+1 ( x ) xP n ( x ) = ( n + 1 ) Pn ( x )
32. ( 1 x ) =
n=0
( )nn!
xn, x (1, 1 ) , R 33. D nx
xk
=
k!
( n k )! xkn, si k > n
0 si k < n
34. Ecuacion de Bessel : x 2 y + xy +
x 2 + 2
y = 0 , R 35. J ( x ) =+n=0
(1 )nn! ( n + 1 )
x2
2n
36. J ( x ) =
+
n=0
(
1 )
n
n! ( n + + 1 ) x
2 2n+
37. Jm ( x ) = (1 )m
Jm ( x ) 38. J1 ( x ) + J+1 ( x ) =
2
x J ( x
39.d
dx
xJ ( x )
= xJ+1 ( x ) 40. d
dx[ x J ( x ) ] = x
J1 ( x ) 41. J1 ( x ) J+1 ( x ) = 2J ( x )
42. Ecuacion Hipergeometrica : x ( 1 x ) y + [ ( + + 1 ) x ] y y = 0 , , , R
43. 2F1 ( , ; ; x ) =+n=0
( )n ( )nn! ( )n
xn 44. 2F1 ( , ; ; x ) = ( )
( ) ( )
+n=0
( + n ) ( + n )
n! ( + n )xn
46. 2F1 ( , ; ; x ) = 2 F1 ( , ; ; x ) 47.d
dx[ 2F1 ( , ; ; x ) ] =
2F1 ( + 1 , + 1 ; + 1 ; x )