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Transcript
Page 1: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

Page 4: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

=

−−

−∞

[ ] [ ] 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 Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

Contoh:

4.1. Fungsi Gamma

dxexlim

dxex)2(

bx1

x

0

12

=

−∞

..........

dxexlim 0

x1

b

=

= ∫−

∞→

Page 8: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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 Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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.

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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

Page 14: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

Page 15: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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(

Page 16: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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=

==

==

∫∫

∫∫∫∞

−∞

∞−

∞−

∞−

Page 18: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

Page 20: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

=

−=

−=

−−

Page 22: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

HUBUNGAN Fungsi Beta dengan Fungsi Gamma

4.2. Fungsi Beta

)n( )m()n,m(B

ΓΓ=

)nm(

)n( )m()n,m(B

ΓΓ=

Page 23: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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=

ΓΓ=

Page 24: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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:

Page 25: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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 Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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

==

−=−

=

−=

−=

∫∫

∫∫∫−

Page 27: Fungsi Gamma Beta Print.ppt - Gunadarma Universitypardede.staff.gunadarma.ac.id/Downloads/files/37636/Fungsi+Gamma... · Tabel Nilai Fungsi Gamma 4.1. Fungsi Gamma n ΓΓΓΓ(n) 1,00

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