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INTEGRAL
EULERIANA |
GAMMA |
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(Integral impropia de
1ª especie) |
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Dependen de un parámetro p>0 y no de la variable
de integración, con la expresión: |
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∫ |
∞ |
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Converge:
p > 0 |
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Γ(p) = |
e-x xp-1dx |
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Diverge...: p = 0 |
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0 |
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PROPIEDADES |
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∫ |
∞ |
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| 1) |
Γ(1) = 1 |
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Γ(1) = |
e-x dx |
= |
-e-x |
∞ |
= 0 - (-e0) = 1 |
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0 |
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0 |
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∫ |
∞ |
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∫ |
∞ |
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| 2) |
Γ(p) = (p-1) Γ(p-1) |
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Γ(p) = |
e-x xp-1dx |
= |
-e-x x p-1 |
∞ |
+ (p-1) |
e-x xp-2dx |
= |
(p-1) Γ(p-1) |
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0 |
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si p>1 , p € R+ |
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Ley de recurrencia |
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xp-1 |
= |
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(p-1)xp-2dx |
= |
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Por partes y |
lim (-e-x x p-1) |
= 0 |
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de la Gamma |
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e-x dx |
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dv |
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-ex |
= |
v |
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x→∞ |
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| 3) |
Γ(p) = |
(p-1)(p-2)
··· (p-n) Γ(p-n) |
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si p>1, p € R+, n € Z+, |
con n el mas próximo y menor que p. |
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Donde siempre 0 <
(p-n) < 1. Existen tablas para Γ(x) donde 0 < x < 1. |
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| 4) |
Γ(n) = |
(n-1)! |
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si n>1, n € Z+ |
Se aplica la propiedad 3 hasta el entero anterior a
n. |
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∫ |
∞ |
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∫ |
∞ |
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x = t1/n |
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∫ |
∞ |
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(p - 1 = 1/n - 1) |
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| 5) |
-xn |
dx |
= 1/n
Γ(1/n) |
si n>1, n € Z+ |
-xn |
dx |
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1/n |
t(1/n)-1e-tdt = |
1/n Γ(1/n) |
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| e |
e |
dx=1/nt1/n -1 |
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0 |
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| 6) |
Γ(1/2) |
= |
√π |
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(Se demostrara en la propiedad 6 de la función
Beta) |
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EJEMPLOS |
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∫ |
∞ |
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(p-1=2) |
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| 1) |
x2 e-xdx = |
Γ(3) = |
2! |
= |
2 |
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0 |
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∫ |
∞ |
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∫ |
∞ |
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(p-1=1) |
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(prop:4) |
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| 2) |
x e-(x+3) dx = |
e-3 |
x e-xdx = |
e-3 Γ(2) = |
e-3 |
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0 |
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0 |
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∫ |
∞ |
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∫ |
∞ |
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(∫ |
∞ |
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-∫ |
∞ |
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| 3) |
1/3(x-3)e-(x-3)dx = |
1/3 e3 |
(x-3)e-xdx = 1/3 e3 |
xe-xdx |
3e-xdx |
) = |
1/3 e3[Γ(2)-3Γ(1)] = -2/3 e3 |
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∫ |
∞ |
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(prop:5) |
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(prop:6) |
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| 4) |
-x2 |
dx |
= |
1/2 Γ(1/2) |
= |
1/2 √π |
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| e |
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0 |
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∫ |
∞ |
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(p-1=3/2) |
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(prop:3) |
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(prop:6) |
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| 5) |
x3/2 e-x dx = |
Γ(5/2) = |
(5/2-1) (5/2-2) Γ(5/2-2) |
= |
3/2 1/2 Γ(1/2) = |
3/4 √π |
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