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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)=!5"#" $ (1/)=%!&5&

(1/%)=!&" $ (/5)=!1

(%/5)=1!" $ (/%)=1!%51

(%/)=1!5 $ (/5)=1!1& !

1$#

1$1

1$1$%

1$

1$5

1$&

1$

1$

1$"

$#

1$####

#$"51

#$"1#$"5

#$%

#$&

#$"%5

#$"#&

#$"%1

#$"&1

1$####

1

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Ifxis an integern= 1, 2, 3, ..., then

*(n) = dxex xn

#

1

*(n+1) = n *(n)

* (,) * (1-,) =

,sin # . , . 1

http://mathworld.wolfram.com/Integer.htmlhttp://mathworld.wolfram.com/Integer.html

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+#

1

d1

px=

,sin # . , . 1

, ,

Contoh &1'%%0)(de!!-

#

% ====

a

%()

1&

1

)(d!

#

'

%

#

'%'/'

'

'

==

=

xx

deeb xx

Contoh. x3 ymisalkan,dxexx3

0

=

= 1/"y d = /"y dy

'

(%)!

#

''/''/' ==

dyey y

Contoh. dxx%-

#

e

misalan y% =x

= y%

d = % ydy

#

'y-'/% dy%yey

1&

%15%

)(%y%

'

1

'

%

'

5

'

"/'

#

/'

==

===

dye y

Contoh. ymisde-

#

=

x

= y

d = y dy

dy'ye)(y-

#

/1'

y

%

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dyye'y-

#

'/1

y = 'dye''

5y-

#

'/%

=

y

='

%

'

1

'

%!'

'

1=

Contoh ymisde%-

#

%

=

= y1/%

d = 1/%y-/%dy

1/%%

1dye

%

1y-

#

%/'

=

y atau *(/%)

Contoh. ymisalande!'-

#

''

=

= 2 y1/

d = 3 y-1/ dy

dyye 1/'-y-

#

1

1

y = dyey-

#

'/1

1&

1

y ='

%

1&

1

=

%'

1

Contoh . dxed'#

#

'ln'-'

= misalan ln = y

= dyy'ln

'

1-

yd

'1

'ln=

'ln

'ln'

1

dy'ln

yed'

'1

-y-

#

-

#

'

=

=

=

Contoh . y-ln misalan!d)/1ln(1

#

% = x

= e-y atau d = - e-y dy

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)%()

%(

dyeyy-!dln

1

%

1

y-

#

%

1#

%

11

#

%

=

==

dyex y

Fungsi Beta.

4ungsi eta dinyataan sebagai beriut :

a . (m $ n) =

>

1

#

#mdannuntuconvergendxx)(1x 1n1m

b ! (m $ n) = (n $ m)

c . =/2

0

1-2m1-2n

n),B(m2

1

dcossin

d .

==+

#

1,

,6sin

6,)-1((,)*d

1

( # . , . 1 )

e .)nm(*

)n(*!)m*()n$m(

+=

Contoh : 7itung =1

#

)(d)-1( 5$"x

&%51

01%00

)1()()(

5"

== =

Contoh : 7itung =1

#

5ymisald)-1(x

= y

d = ydy

=1

#

1

#

5"5 )1(y''ydy)y-1( dyyy

( 1# $ &) =)(

)()('1&

&1#

=15#15

1

Contoh : 7itung =1

#

%1/'-%ymisaland)-1(

5

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= y1/%

d =%

1y - /%dy

=1

#

1

#

'

1-

%

'-%

'-

'

1-

dy)y-1(y%

dy%

y)y-1(

1

)&(

)'()

%(

)%(

5

11

%

1

'

1$

1

%

1

=

Contoh : 7itung

'ymisaland-'

'

#

'

=

d = dy

=1

#

'

1-

1

#

''

)y-1(y'

'y-'

dy'('y)dy

15

'&

)'

(

)'

1((%)

'

)

'

1$%(

'

=

=B

Contoh : Hitung

%ymisalan-%

dd

-%

1%

#

%

#'

==

d = % dy

)'()

'(

)'(dy%)%y-%()%(

11

'

1$

11

#

'

1-

'

1

==

= y

Contoh : 7itung ='

#

'%/''ymisaland)-(

= y1/

d = y-1/ dy

&

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1

#

1/'-%/'dyy)y-(

1

#

1/'-%/'%/' dyy)y-1(

= ( 2 $ 5/ ) =

%'

)%(

)5/'()'/1(

'

1

'

%

==

Contoh +=

%

%ymisaland)-()%-(

d = dy

+1

#

1/1/ dy)%-y-()%-%y(

=1

#

1/1/ )(dy)y-()y(

5$

5

))((%

'

)'/5(

)/1()1/(

'1/

1

1=

=1

#

1

#

' dy)y-1(y%dyy%)y-1( '

%5

1

)(

)5((%)%5)$%(% =

=

Contoh : 7itung =1#

#

'' 1##ymisaland-1##

= 1# '1

y

d = 5 '1

y dy

=1

#

1

#

'

1

'

1-

'

1-

'

1

dy)y-1(y5#dyy5)y-11#(

'5)'(

)'

1()

'

1(

'

5#)

'

1$

'

%(5# =

=

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Contoh 7itung '/

#

dctg

= ='/

#

1/'-1/' )('

1d)(sin)cos(

1$

%

1n

'

11'

%m

'11'

==

==

n

m

)('

1

1$

%=

'

)(

)()

(

'

1

1

1%

=

=

oa! " soa!

1. 7itung =1

#

'n

/1nsinmisalan

)-1(

d

n

2. ='

#

%% %ymisaland-

3. ='/

#

/'

#

1/'-dcos

cos

d

#. ='/

#

/'

#

1/'-1/' dcossindtg

$. 7itung

=+

#

ymisalan

1

d

%. utian !)(

)()(')-1()1( 18,1-81-,

1

1 qp

qpdx

+

=+ ++

&. 9arilah

'

%!:;bde

#

-x

'. 9arilah'm

!:;b#

'' dxe xm

(. 9arilah1&

'!:;be

''-

#

' dxx

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1). 9arilah%%

!:;b1

#

dx

x

x

+

*+I- /IH

"