1
Pertemuan VII
Fungsi Gamma dan Fungsi Beta
Fungsi Gamma
Euler's gamma function is defined by the integral
Some further values of the Gamma function for small arguments are:
(1/5)=4.5909 , (1/4)=3.6256
(1/3)=2.6789 , (2/5)=2.2182
(3/5)=1.4892 , (2/3)=1.3541
(3/4)=1.2254 , (4/5)=1.1642.
1,01,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
2,0 / 1,0000
0,9514
0,9182
0,9875
0,8873
0,8862
0,8935
0,9086
0,9314
0,9618
1,0000 /
If x is an integern = 1, 2, 3, ..., then
Γ(n) =
Γ(n+1) = n Γ(n)
Γ (p) Γ (1-p) = 0 < p < 1
= 0 < p < 1
, ,
Contoh
Contoh.
x = 1/9 y2 dx = 2/9 y dy
Contoh. misalkan
x = y3
dx = 3 y2 dy
Contoh.
x = y2
dx = 2y dy
=
=
Contoh
x = y1/3
dx = 1/3 y-2/3 dy
1/3atau Γ(4/3)
Contoh.
x = ½ y1/2
dx = ¼ y-1/2 dy
= =
=
Contoh . misalkan 4 x2 ln 2 = y
x =
Contoh .
x = e-y atau dx = - e-y dy
Fungsi Beta.
Fungsi Beta dinyatakan sebagai berikut :
a . B(m , n) =
b . B(m , n) = B(n , m)
c .
d . ( 0 < p < 1 )
e .
Contoh : Hitung
Contoh : Hitung
x = y2
dx = 2ydy
2 B ( 10 , 6 ) =
=
Contoh : Hitung
x = y1/3
dx = y - 2/3 dy
Contoh : Hitung
dx = 2 dy
Contoh : Hitung
dx = 3 dy
Contoh : Hitung
x = 2y1/2
dx = y-1/2 dy
= 8 B ( ½ , 5/2 ) = 8
Contoh
dx = 4 dy
Contoh : Hitung
x = 10
dx = 5 dy
Contoh Hitung
=
Soal - soal
1. Hitung
2.
3.
4.
5. Hitung
6. Buktikan
7. Carilah
8. Carilah
9. Carilah
10. Carilah