Fungsi Gamma Beta Print.ppt - Gunadarma...
Transcript of Fungsi Gamma Beta Print.ppt - Gunadarma...
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KALKULUS 4
Dr. D. L. Crispina Pardede, SSi., DEA.
SARMAG TEKNIK MESIN
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1. Deret Fourier
1.1. Fungsi Periodik
1.2. Fungsi Genap dan Ganjil,1.3. Deret Trigonometri,1.4. Bentuk umum Deret Fourier,1.5. Kondisi Dirichlet,1.6. Deret Fourier sinus atau cosinus separuh jangkauan.
2. Integral Fourier
KALKULUS 4 - SILABUS
2. Integral Fourier
3. Transformasi Laplace
3.1. Definisi dan sifat Transformasi Laplace3.2. Invers dari transformasi Laplace3.3. Teorema Konvolusi3.4. Penerapan transformasi Laplace dalam penyelesaian P. D.
dengan syarat batas.4. Fungsi Gamma dan Fungsi Beta
4.1. Fungsi Gamma
4.2. Fungsi Beta4.3. Penerapan fungsi Gamma dan fungsi Beta
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Tabel Nilai Fungsi Gamma
4.1. Fungsi Gamma
n ΓΓΓΓ(n)
1,00 1,0000
1,10 0,9514
1,20 0,9182 1,20 0,9182
1,30 0,8975
1,40 0,8873 1,50 0,8862
1,60 0,8935
1,70 0,9086
1,80 0,9314
1,90 0,9618 2,00 1,0000
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Grafik Fungsi Gamma
4.1. Fungsi Gamma
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FUNGSI GAMMA : ΓΓΓΓ(n)
4.1. FUNGSI GAMMA
∫∫−−−
∞−
==Γ
bx1nx1ndxexlimdxex)n(
konvergen untuk n>0
∫∫ ∞→==Γ
0b
0
dxexlimdxex)n(
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Contoh:
4.1. Fungsi Gamma
dxexlim
dxex)1(
bx11
x
0
11
=
=Γ
−−
−∞
−
∫
∫
[ ] [ ] 1 ee lim e lim
dxelim
dxexlim
0b
b
b
0
x
b
b
0
x
b
0
x11
b
=+−=−=
=
=
−
∞→
−
∞→
−
∞→
−−
∞→
∫
∫
![Page 7: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/7.jpg)
Contoh:
4.1. Fungsi Gamma
dxexlim
dxex)2(
bx1
x
0
12
=
=Γ
∫
∫
−
−∞
−
..........
dxexlim 0
x1
b
=
= ∫−
∞→
![Page 8: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/8.jpg)
Rumus Rekursi dari Fungsi Gamma
ΓΓΓΓ(n+1) = n ΓΓΓΓ (n)
dimana Γ(1) = 1
4.1. Fungsi Gamma
Contoh:1. Γ(2) = Γ(1+1) = 1 Γ(1) = 1.
2. Γ(3) = Γ(2+1) = 2 Γ(2) = 2.
3. Γ(3/2) = Γ( ½ +1) = ½ Γ(½).
![Page 9: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/9.jpg)
Bila n bilangan bulat positif
ΓΓΓΓ(n+1) = n!
dimana Γ(1) = 1
4.1. Fungsi Gamma
Contoh:1. Γ(2) = Γ(1+1) = 1! = 1.
2. Γ(3) = Γ(2+1) = 2! = 2.
3. Γ(4) = Γ(3+1) = 3! = 6.
![Page 10: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/10.jpg)
Contoh:Hitunglah
4. Γ(6)
5. Γ(5)/Γ
4.1. Fungsi Gamma
5. Γ(5)/Γ(3)
6. Γ(6)/2Γ(3)
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Bila n bilangan pecahan positif
ΓΓΓΓ(n) = (n-1) . (n-2) . … αααα ΓΓΓΓ(αααα )
dimana 0 < α < 1
4.1. Fungsi Gamma
Contoh:1. Γ(3/2) = (1/2) Γ (1/2)
2. Γ(7/2) = (5/2)(3/2)(1/2)Γ(1/2)
3. Γ(5/3) = (2/3)Γ(2/3).
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Bila n bilangan pecahan negatif
4.1. Fungsi Gamma
n
)1n()n(
+Γ=Γ
atau
)...1n(n
)mn()n(
−
+Γ=Γ
m bilangan
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Contoh:
4.1. Fungsi Gamma
−
−
+−Γ
=
−
−Γ
=
−
+−Γ
=
−Γ
13
12
1
3
2
1
3
12
3
2
3
Γ
=
−
−
Γ
=
−
−−−
4
3
2
1
2
1
2
3
2
1
2
1
2
3
2
3
2
32
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Contoh:
4.1. Fungsi Gamma
−
−
+−Γ
=
−
−Γ
=
−
+−Γ
=
−Γ
11
2
3
2
5
12
3
2
5
2
3
2
5
12
5
2
5
−
Γ
=
−
−
−
Γ
=
−
−
−
+−Γ
=
−
−
−Γ
=
8
15
2
1
2
1
2
3
2
5
2
1
2
1
2
3
2
5
12
1
2
3
2
5
2
1
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Beberapa hubungan dalam fungsi gamma
4.1. Fungsi Gamma
π=Γ )( 21
)!1n()n( −=Γ )!1n()n( −=Γ
n
)1n()n(
+Γ=Γ
π
π=−ΓΓ
nsin)n1()n(
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Contoh Soal:
4.1. Fungsi Gamma
( )( )
( ) ( )
21
25
.3Γ
Γ( )
( )1 .2
2
5 .1
−Γ
Γ
( ) ( )
( )
( )( )
325
386
.5
5,5
5,23 .4
Γ
Γ
Γ
ΓΓ( )
( )( )
21
21
.3
21 .2
Γ
−Γ
−Γ
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Penggunaan Fungsi Gamma
==
=→=
0y maka 0, x bila
dy dx y 2x Misalkan
:awabJ
21
4.1. Fungsi Gamma
∫∞
−
0
x26dxex Hitung .1
∞=∞=
==
y maka , x bila
0y maka 0, x bila
8 45
2
6! (7) )(
dyey )( dyey )(
dyey)( dye)y( dxex
7
7
21
0
y1-77
21
0
y67
21
0
y67
21
0
21
y6
21
0
x26
==Γ=
==
==
∫∫
∫∫∫∞
−∞
−
∞−
∞−
∞−
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4.1. Fungsi Gamma
∫∞
−=
0
x y substitusidengan dye y Hitung .233y
∞=∞=
==
=→=
y maka , x bila
0y maka 0, x bila
dy 3y dx x y Misalkan :awabJ23
π=Γ==
==
=
=
∫
∫∫
∫∫∫
∞−
∞−
∞−
∞∞∞−
3
1)(
3
1dx e x
3
1
dx e x3
1 dx e x
3
1
dxx 3
1 e xdx
x 3
1 e x dye y
21
x-
0
121
x-
0
21
x-
0
32
61
x-
0
61
23
1
x-
0
31
03
2
3y
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4.1. Fungsi Gamma
u ln x substitusidengan
ln x-
dx
Hitung .3
1
0
=−∫
0 u maka ,1 x bila dan u maka 0, x Bila
due- xd e x u ln x Misalkan
:awabJu-u-
==∞==
=→=→=−
0 u maka ,1 x bila dan u maka 0, x Bila ==∞==
π=Γ==
−==
∫
∫∫∫∞
∞∞
−
−
)( du e u
du )e( u u
due-
ln x-
dx
21
0
001
0
u-
u--u
21
21
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FUNGSI BETA dinyatakan sebagai
4.2. FUNGSI BETA
∫−−
−=
1
0
1n1mdx)x1(x)n,m(B
konvergen untuk m > 0 dan n > 0.
Sifat: B(m,n) = B(n,m)
Bukti: … … …
0
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Bukti:
4.2. Fungsi Beta
dx)y()y1(
dx)x1(x)n,m(B
11n1m
1
0
1n1m
−=
−=
∫
∫
−−
−−
Terbukti
)m,n(B
dx)y1()y(
dx)y()y1(
1
0
1m1n
0
∴
=
−=
−=
∫
∫
−−
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HUBUNGAN Fungsi Beta dengan Fungsi Gamma
4.2. Fungsi Beta
)n( )m()n,m(B
ΓΓ=
)nm(
)n( )m()n,m(B
+Γ
ΓΓ=
![Page 23: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/23.jpg)
Contoh:1. Hitung B(3,5).Jawab:
4.2. Fungsi Beta
)5( )3()5( )3( ΓΓΓΓ......
)8(
)5( )3(
)53(
)5( )3()5,3(B =
Γ
ΓΓ=
+Γ
ΓΓ=
![Page 24: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/24.jpg)
Contoh:2. Hitung B(5 , 2).
Jawab:3. Hitung B(3/ , 2).
4.2. Fungsi Beta
3. Hitung B(3/2 , 2).
Jawab:4. Hitung B(1/3 , 2/3).
Jawab:
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Penggunaan Fungsi Beta
4.2. Fungsi Beta
∫ −
1
0
34dx)x1(x Hitung .1
:Jawab11
280
1
!8
!3 !4
4) (5
(4) (5)
)4,5(B
dx)x1(x dx)x1(x
1
0
141-51
0
34
=
=+Γ
ΓΓ=
=
−=− ∫∫−
![Page 26: Fungsi Gamma Beta Print.ppt - Gunadarma Universityfeni.andriani.staff.gunadarma.ac.id/Downloads/... · Fungsi Periodik 1.2. Fungsi Genap dan Ganjil, 1.3. Deret Trigonometri, 1.4.](https://reader034.fdokument.com/reader034/viewer/2022042606/5f7ea5ddfdca6f57a04b882c/html5/thumbnails/26.jpg)
4.2. Fungsi Beta
∫−
2
0
2
x2
dxx Hitung .2
du 2 dx 2u x Misalkan :Jawab =→=
...)(3,B 24
du)u1(u 24 u1
duu
2
8
u12
duu8
u22
du2)u2(
x2
dxx
21
1
0
21
0
2
1
0
21
0
22
0
2
21
==
−=−
=
−=
−=
−
∫∫
∫∫∫−
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Penggunaan Fungsi Beta
4.2. Fungsi Beta
dy yay Hitung .3
a
0
224
∫ −
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LATIHAN
dx e x Hitung .3
dx e x Hitung .2
x4
0
x4
∫
∫∞
∞−
−
)(
)()3( b).
)3()4(2
)7( a). Hitung .1
29
23
Γ
ΓΓ
ΓΓ
Γ
dx e x Hitung .3
0
x4∫−
du u)-4( u Hitung .6
dx )x1( x Hitung .5
42/53/2
132
0
0
∫
∫ −
)32,3
12
3 (B b). 2) ,(B a). Hitung .4