fungsi gamma, fungsi beta, integral...
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Fungsi Gamma, Fungsi Beta,Integral Dirichlet
Kalkulus 3Teknik Industri
Universitas Gunadarma
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FUNGSI GAMMA
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1 2 3( ) lim , 0, 1, 2,( 1)( 2) ( )n
znz n zz z z z n
v Adalah generalisasi dari fungsi faktorial n! untuk nilai non-integer.
Definisi # 1
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1 2 3( ) lim , 0, 1, 2,( 1)( 2) ( )
1 2 3( 1) lim ( ) lim( 1)( 2) ( 1) ( 1)
( 1) ( )
1 2 3(1) lim
n
n n
n
z
z
nz n zz z z z n
n nzz n zz z z n z n
z z z
n
Note that 1 2 3 n
1 (2) 1 (1) 1,( 1)
(3) 2 (2) 2 1, (4) 3 (3) 3 2 1, (5) 4 (4) 4 3 2 1, .
nn
etc
, and
4
Sifat faktorial:
( ) 1 !n n ( 1) !n n atauMaka
Perhatikan dan
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Disebut juga definisi bentuk integral Euler
Leonard Euler
ln ln1 1 1 1
1 1 0
iyiy t iy tz x x x
z x
t t t t e t e
t t x
for integral to converge
Catatan:
0
1( ) , Re 0t zz e t dt z
Definisi # 2
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0
0
1
0
2
1
2 1 2
1
( ) , Re 0
( ) 2 , Re 0
1( ) ln , Re 0 ln 1 /
t z
s z
z
z e t dt z
z e s ds z t s
z ds z t ss
The following three integral definitions are all equivalent :
(let )
(let )
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Ketiga definisi berikut adalah ekivalen
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1
1 1( )
zz n
n
zze ez n
7
Definition # 3
The Weierstrass product form can be shown to be equivalent to definitions #1 and #1.
0.5772156619 where is the Euler -Mascheroni constant.
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( ) (1 )sin
z zz
Tetapkan z = 1/2:
(1 / 2)
Bentuk yang sering digunakan adalah (1/2).
Gunakan formula di atas:
( ) (1 )sin
z zz
Formula Refleksi Euler
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9Note: There are simple poles at z = 0, -1, -2,… 1
Res!
n
nn
xy
z
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(x) and 1 / (x)
Oleh karena (z) tidak bernilai 0 maka (1 / (z) adalah analitik di mana-mana).
Catatan : (x) tidak bernilai 0.
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Formula Sterling (deret asimtotis untuk z besar):
2 3 4
1 1 1 139 5712 112 288 51840 2488320
z zz z ez z z zz
, argz z constant
Berlaku untuk
3 5
1 1 1 1ln ln ln2 2 12 360 1260
zz z z zz z z
Dengan me-ln-kan kedua ruas, diperoleh:
2 3
ln 12 3w ww w Note :
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! 1 2 3 2 1n n n n
0
! 1 t xx x e t dt
0
! 1 t zz z e t dt
Integer
Bilangan real
Bilangan kompleks
1x
Re 1z
Rangkuman generalisasi faktorial
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0
! 1 t zz z e t dt
Bilangan kompleks
Re 1z
0
! 1
( ) (1 )sin
t zz z e t dt
z zz
Bilangan kompleks0, 1, 2z
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Fungsi Gamma (1)
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Fungsi Gamma (1)
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Fungsi Gamma (2)
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Fungsi Gamma (2)
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FUNGSI BETA
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Fungsi Beta
1 1 1
0( , ) (1 )p qp q x x dx
p dan q disebut parameter bentuk.
( , ) ( , )p q q p
( ) ( )( , )( )p qp qp q
Pada x = 0, suku ini…• bernilai 0 jika p > 1• memiliki sebuah singularitas jika 0 < p < 1.
Pada x = 1, suku ini…• bernilai 0 jika q > 1• memiliki sebuah singularitas jika 0 < q < 1.
0 1
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• Dengan menggunakan transformasi x=sin2 θ diperoleh:
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Fungsi Beta (1)
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Fungsi Beta (1)
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Fungsi Beta (2)
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Fungsi Beta (2)
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Fungsi Beta (3)
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Fungsi Beta (3)
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INTEGRAL DIRICHLET
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Definisi
Jika V menyatakan daerah tertutup ,x ≥ 0,y ≥ 0, z ≥ 0, di mana a>0, b>0, c>0, p>0, q>0, r>0maka:
Integral di atas disebut integral Dirichlet(Kasus khusus: a=b=c=p=q=r=1)
≤ 1
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