اـماـج ةـلاد لوـح -...
Transcript of اـماـج ةـلاد لوـح -...
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2016مارس 15السنة العاشرة العدد مجلة الساتل
61
حـول دالـة جـامـا
The Gamma function (1-1)
z
21
1.2.1
nzzzz
znn
nLimz
21
!1
0z
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حـــول دالـــة جـامــا
62
1 z0zz
0
1
z
z,2,1,0 z
1.2.1
11
1
n
n
nLim
1z z1
zzzz
nz
zn
nLim
nzzzz
znn
nLim
nz
zn
nzzzz
znn
nLim
nzzz
znn
nLimz
121
!
121
!
121
1!1
zzz 12
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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!114321
21222123
111112
nnn
!13 nn
Nn
1.2.2
0
14 dtzttez
nn
t
nLimte
1
1.2.2
ndtzt
n
n
t
nLimz
0
115
n
ty10 y
1
0
11 dyn
zny
ny
nLimz
n
nzzz
znn
nLimz
1
!z > 0
1.2.1
1
2
1
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حـــول دالـــة جـامــا
64
1.2.2
0
2
1
0
12
1
2
1
dttte
dttte
2xtdxxdt 2
0
22
2
1dxxe
0
2dxxeI
0 0
222 dydx
yxeI
2
00
2
2
1
2
0 0
22
dre
ddrrreI
42
12
1
2
I
22
2
1
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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2
2
1
2
1
)1()(
)()1(
z
z
zz
zzz
2
2
1
)2
1(
)2
1(
2
1
)12
1(
)2
1(
3
0
3. dxxex
2)
3
1
(23
1
23
23
3
0
1)()(
yxyx
x
dydx
dxxdy
xyأننفرض
dtzttez
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حـــول دالـــة جـامــا
66
0.2
1
3
1
0
3
2
6
1
3
1
3
2
3
..2
1
)0
3
1
(0
3.2
1
dyy
ey
dyy
ey
y
dyyeydxxex
3)
2
1(
3
13
.0
0
2
1
.3
1)
2
1(
3
1
2
1
2
11
0
1.)(
dxx
ex
dyyy
e
zzأننجد
dyzyy
ez
1.2.1
zzz
sin)1().(
)2()1...()3()2()1(
)1(!
)1(
1
)1()...()2()1(
!)(
znzzz
znn
nLimz
zzأننفرض
nzzzz
znn
nLimz
12
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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)1)...(2)(1)(...()2()1(
)1(!.!
)1()(znzznzzzz
znnznn
nLimzz
2)!(
)1)(22...()223()222()221(
.2)!(
2)!(
1
)1)(22...()223()222()221(
.2)!(1
n
znznzzz
nn
n
nLim
z
znznzzz
nn
nLim
z
)2)(1...()2)3
(1()2)2
(1()2)1
(1(
11
1lim.
)2)(1...()2)3
(1()2)2
(1()2)1
(1(
11
)1)(2)(1...()2)3
(1()2)2
(1()2)1
(1(
.1
n
zzzznLim
z
zn
n
n
zzzznLim
z
znn
zzzz
n
nLim
z
zzz
n
znzzz
sin)1().(
2)(1
1
1
1)1().(
4
nnn )1()2
1()
2
1(
nzنضع
zzz
2
1
sin)1()(
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حـــول دالـــة جـامــا
68
)2
1sin(
)2
1()
2
1(
)2
1sin(
))2
1(1((.
2
1)
nnn
nnى
yxyxyx sincoscossin)sin(
nn
nn
nn
)1()1(
)2
1()
2
1(
)1()2
1sin(
1.2.2
1)(
)(lim
nz
nzn
n
)...()2()1(
!)(
nzzzz
znn
nLimz
)...()2()1(
).(
nzzzz
znnn
nLim
zzznznzn
nznznn
nLim
)1...(11
)1(...)(
)...2)(1).()(1)...(1)((
)...2)(1(..).(
zzzznznz
zzznznn
nLim
)2)(1)...(1)((
.).(
zznznz
nznn
nLimz
)1(
.).()(
zn
nznn
nLimzz
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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1)(
).(
1.)(
).(1
lim)(
).(1
)()(
).(.
zn
znn
nLim
zn
znn
nLim
zn
n
nzn
znn
nLim
znzn
znnn
nLim
1.2.3
Nn
)1)(3)...(32)(12(2
1)
2
1( nn
nn
)1()1()( xxx
)12(2
1)
2
1( nn
12
121
2
12
2
12
nnn
)1)(3)...(52)(32)(12(
2
1)
2
1(
)2
1()1)(3...(
2
52
2
32
2
12
5
52
2
52
2
32
2
12
2
32
2
32
2
12
2
12
2
12
2
1
nnnn
n
nnn
nnnn
nnn
nnn
1.2.4
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حـــول دالـــة جـامــا
70
Nn
)1)(3)...(52)(32(12
1)
2
1(
nn
nn
)2
1( n
)1()12
1()
2
1( mnn
)2
1()
2
1(
)(
nn
mm
2
1
2
12
n
n
1.2.3
n
nnn
nn
n
2
1)1)(3)...(32)(12(
12
2
)2
1(
12
2)
2
1(
)1)(3)...(52)(32(12
1)
2
1(
nn
nn
5
Nn
)1)(3)...(32)(12)(12(12
1)
2
3(
nnn
nn
)2
3( n
)2
1(
2
12
)2
1()
2
1(
)(
)1()12
1()
2
3(
nn
nn
mm
mnn
1.2.3
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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)1)(3)...(12)(12(12
1
2
1)1)(3)...(32)(12(
2
12)
2
3(
nnn
nnn
nn
6
)2
1()!(
22)!2( nn
nn
)2
1)(
2
3)(
2
5()
2
3)(
2
1.(123)...3)(2)(1(.
22
)2
1(2).1(2).
2
3(2).2(2).
2
5(2)...2(2).
2
3(2).1(2).
2
1(2).(2
12345)...32)(22)(12.(2)!2(
nnnnnnnn
nnnnn
nnnnn
)2
1().!.(
22)!2(
)2
1(
)2
1(!..22
)...2
5)(
2
3)(
2
1(
...2
5.
2
3.
2
1.
2
1.
2
3)...
2
3)(
2
1(!.22
nn
n
n
nnn
nnnn
7
(8) !
12345678!8
n28
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حـــول دالـــة جـامــا
72
)2
9()!4.(
82)!42(
824
)2
1(!..
22)!2(
42
8
nn
nnn
n
n
3571234
42
)2
1(
42
13571234
42.42
)2
1(.
2
1.
2
3.
2
5.
2
7.1234.
82)!8(
!812345678
321234567
35712344
2
1.2.3
n
k
nkn
Lim
n
x
en
x
n
xex
x
1
ln1
11
16
0.577216
1.2.11.2.3
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2016مارس 15السنة العاشرة العدد مجلة الساتل
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k
xn
k
xn
nLimx
n
xxx
xnnLimx
n
nxxx
xnnLimx
xnn
nxxx
nLim
x
11
12
11
11
321
211
!
11
nxexn
ln
nxe
nLim
xe
nxe
nn
x
enLim
nx
enLim
n
x
e
x
ex
enLim
k
x
ek
ln
lnln
1
2
11
1
2
11
2
1
k
x
en
knLim
xexn
nLim
17
7
n
x
en
x
n
xex
k
xk
x
en
knLim
xex
k
xn
knLimk
x
en
knLim
xex
x
11
11
111
1
1.2.11.2.3
8
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حـــول دالـــة جـامــا
74
22
21
1sin
n
x
nxx
xx
xx
sin
1.2.3
n
x
en
x
n
xex
x
1
1
1
2
21
1
2
111
21
n
x
nx
n
x
en
xn
x
en
x
nx
xx
xx
2
21
1ins
n
x
nxx
x
xx
xx
sin21
xx
xx
sin8
x
9
0
)(1222 zdtztte
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2016مارس 15السنة العاشرة العدد مجلة الساتل
75
2trr0dttdr 2
)(0
1.
.2.0
222
.)2(2
0
22
0
1222
zdrzrre
dttztte
dttz
tte
dtztte
2
0
12sin.12cos2
)(
)().(
dyx
yx
yx
0
1222)( dtxttex
0
1212
0
)22(4)().(
0
1222)(
dtdsy
sxtst
eyx
dsy
ssey
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حـــول دالـــة جـامــا
76
2
0
12sin12cos2
)(
)()(
2
0
)(12
sin.12cos2)().(
2
0
1)(2
0
22.
12sin12cos2
2
0
12sin.12cos.
0
1)(2.
24
2
0
.12
)sin.(
0
12)cos.()2sin22cos2(
4)().(
20,0/sin,cos(
0,0/),(
dyx
yx
yx
yxdyxyx
dryx
rredyx
ddryxyx
rre
drdry
rxrrr
eyx
rrr
or
stst
1.2.4
)2
1().(.122
)2(
zzz
z
10
2log2
)2
1(
)2
1(
)(
)(
)2(
)2(2
z
z
z
z
z
z
)2
1().(.122
)2(
zzz
z
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2016مارس 15السنة العاشرة العدد مجلة الساتل
77
log)2
1(log)(log2log)12()2(log zzzz
z
2log2)
2
1(
)2
1(
)(
)(
)2(
)2(2
0)2
1(
)2
1(
1)(
)(
12log2)2(
)2(
2
z
z
z
z
z
z
zz
zz
zz
)2log2()2/1(
2log2
)2
1(
)2
1(
)(
)(
)2(
)2(2
z
z
z
z
z
z
Z=1/2
)2log2()2/1(
2log2)2/1(
2log2)2/1(
2log2)2/1(
)1(
2log2)1()2/1(
)1(2
2log2)1(
)1(
)2/1(
)2/1(
)1(
)1(2
11
)12...(5.3.1
2..)1()(
m
mmz
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حـــول دالـــة جـامــا
78
))1()()(1)...(2)(1()(
)()1)...(3)(2)(1()(
mzmmzmmzmmzmzmz
zzmzmzmzmz
)...2)(1)()(1)...(2)(1(
)...1())(1)...(2)(1(
zzzzzmzm
mzmmzmmzmzmzm
)(.).1)...(3)(2)(1()(
)())(1)...(2)(1(
zzzmzmzmzmz
zzzzmzm
)).(1)...(2)(1(
)()(
zzmzmz
mzz
12
11,
2
1
2
1
mzmz
mz
)12...(5.3.1
.2.)1()(
2
)1(
2
1
)12)...(7)(5)(3)(1(2
1
)12(2
1)32(
2
1...
2
5
2
3
2
1
)2/1()(
m
mm
z
m
mm
mm
mmz
1.2.5
Nn
).
1(.
1
0mn
m
ndxm x
nxe
dxn
nxdyn
xy1
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2016مارس 15السنة العاشرة العدد مجلة الساتل
79
dxmxnxedxm x
nxe
nyx
nnx
dydx
1
00
1
1
dynnmyy
en
dynmnyy
en
dyn
nymnyy
en
nnyn
dymnyy
e
0
111
.1
11
.
1
.
0
1
.)1(
1
.
0
.
1
.1
1)
1
(
.
1
)
1
.(
0
dynm
m
yy
en
11
.
0
1
)(z
)1
(1
)(1
1.
0
111
.
0
1
1
nm
m
nz
n
dyzyy
en
dynm
m
yy
en
nm
mz
12
dxxxe 3
0
2
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حـــول دالـــة جـامــا
80
)1
(1
0nm
m
ndxm x
nxe
)3
2(
2
1)
6
4(
2
13
0
2
3,2
dxxxe
mn
13
!353417
0
)34
1(
dxxxe
)1
(1
0nm
m
ndxm x
nxe
!3534
)36(34)
2
1
18(34
)
34
17
18(34
)
)17()34
1(
117(
34
1
117
0
)34
1(
17,34
1
dxxxe
mn
14
!3984
1
0
8
1
dxxxe
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2016مارس 15السنة العاشرة العدد مجلة الساتل
81
)1
(1
0nm
m
ndxm x
nxe
!398)40(8
)32()4
5(8)
32
14
5
(8
)
)4
1()
8
1(
14
1
(
8
1
14
1
0
8
1
4
1,
8
1
dxxxe
mn
15
)1
(
)1
(
0c
a
cb
c
a
dtcbteat
b,aa>-1
![Page 22: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/22.jpg)
حـــول دالـــة جـامــا
82
cb
cbx
ct
c
a
b
c
a
xat
c
c
b
c
c
xct
cb
cxt
cbtx
1
1)1
(
1
1
)1(
)1(
1,1
1
dx
ccb
cx
dxctcb
dt
dtccbtdx
1
)1(
1
)1(
1
1
0 1
1
1
.0
00
dx
ccb
cx
c
a
b
xec
a
xdt
cbteat
xt
xt
dxxec
a
xc
a
cb
.
0
1)1
()1(
1
dtxttex 1
0
)(
![Page 23: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/23.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
83
)1
(.)
1(
1
0
)1
(0
1)1
(
c
a
c
a
cb
dtcbteat
c
adx
xec
a
x
16
np
1)(log
!1
0
npe
ndtpnt
dt
te
entdtnt
tete
e
)(log
0
1
0
)(log)
)log(
1
a=n)(log,1 e
bc
1)(log
!
1)(log
)1(1
0
ne
n
ne
ndtpnt
31Digamma
x
xx
dx
dx
ln
,2,1,0 x
x
17
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حـــول دالـــة جـامــا
84
0 1
1
n xnn
xx
1.2.3
n
x
en
x
n
xex
x
1
1
1
1
1lnlnln
n n
x
n
xxxx
n
x
n
x
n
xxx
1ln
1
lnln
x
1
111
n nxnxx
x
0 1
1
0
1
1
1
n xnn
x
n xnnx
18
n
nnn
sin!!
x
xx
nnnn
sin)1()(
!)()1(
x=n
![Page 25: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/25.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
85
n
nnsin
)1()(
n
n
nnn
n
nnnn
sin)!(!
sin)1()(
2.1The beta function
1771
2.1.1
0,0 xy
1
0
111,9 dt
ytxtyx
y , x
1
0
11
1, dt
xt
ytxy
ut1dudt
duy
uxu
duxuy
uxy
1)1(
1
0
1
1
0
11)1(,
2.1.1
yx ,
xyyx ,,
2.1.1
1
0
111, dt
ytxtyx
2sint2
0
![Page 26: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/26.jpg)
حـــول دالـــة جـامــا
86
ddt cossin2
0t01t2
dyxyx
2
0
12sin12cos2,
yx
yxd
yx
2
0
12sin12cos2
yx
yxyx
,10
19
0 1
1, dt
yxt
xtyx
0,0 xy
1
0
111, dt
ytxtyx
0,0 xy
u
ut
1
t
tu
1
t01u0
2
1 u
dudt
ut
1
11
![Page 27: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/27.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
87
1
0
111, dt
ytxtyx
0 1
1
0 21
1
1
11
1,
duyx
u
xu
u
duy
u
x
u
uyx
0 1
1, dt
yxt
xtyx
2.1.1
pqحيث
dxqn x
px
nqnp
0)Re(,0)Re(
1)1(
1
0
11),(
,...2,1n
dxn
n
xn
dy
nyxy
nx
1
1
1
![Page 28: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/28.jpg)
حـــول دالـــة جـامــا
88
dyq
ynp
yn
dynyq
ynnp
yn
n
n
nyn
dyqy
pnydxqn x
px
yx
yx
1)1(
1
0
1
1)1(
1
0
)1(
)(1
1)1(
1
0
1)(
1)1(
1
0
1
11
00
)(
)()(
),(1
)1(
1
0
11
qnp
qnp
qnpndxqn x
px
n
2.1.2
)2(
)1()1(1)()()(
nm
nmnmabdxb
a
nxbmax
bamn ,1,
)1()(
)(
)(
)(
)()(
tab
atabbab
dtabdx
atabx
tabax
![Page 29: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/29.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
89
)2(
)1()1(.
1)(
)1,1(1)(
1
0
)1(1)(
)1(1
0
1)(
)()(
1
0
nm
nmnmab
nmnmab
dtntmtnmab
dtntmtnmab
b
a
dxnxbmax
ybx
yax
20
pp
pppp
.14
2
)2
1,
2
1().,(
2)2(2
2)
2
1(.
2)(
)12(
2)
2
1(
.)2(
2)(
)2
1,
2
1().,(
pp
pp
p
p
p
ppppp
ppp
p
pppp
.4
224
2.2
)2
1,
2
1().,(
21
)1,(),1(),( bababa
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حـــول دالـــة جـامــا
90
)2(...),(
)()(
)()(
)1(
)1()()1,(
)1(...),(
)()(
)()(
)1(
)()1(),1(
baba
b
baba
bba
ba
baba
baba
a
baba
baa
ba
baba
12
),(),(
),(),()1,(),1(
baba
baba
baba
bba
ba
ababa
13(The Hypar Geometric series)
)11(....2
)1(!2.1
)1()1(.
!11
xx
0
....21
n
xxnx
1x
1x01x
11
)12(...)(
)(
)1)...(1()(
r
rr
rx
r rr
rrxF .
0 )(!
)()(),,,(
12
![Page 31: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/31.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
91
21
),,,(12
),,,(12
xFxF
12 F
0 1)(!
1)(
1)(
1.
1 )()!1(
)()()),,,(12(
r rr
rr
rx
r rr
rrxFdx
d
),1,1,1(12
.
0 )1(!
)1()1()),,,(
12(
)1(
))...(1()(
)1(
1)(
xF
rx
r rr
rrxF
dx
d
r
rr
r
1)0,,(12
F
0
)),,,(12
(x
xFdx
d
dtxttxF
)1.(
1
0
1)1(
),(
1),,,(
12
x=1
),(),(
1
1
0
1)1(
),(
1)1,,,(12
dttF
![Page 32: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/32.jpg)
حـــول دالـــة جـامــا
92
0,0
)().(
)()()1,,,(
12
F
23
Gauss's thaery
)()(
),()1,,,(
12
F
0,0
Kummar's thaery
)2
11()1(
)2
11()1(
)1,1,,(12
F
1 x=1
),,,(12
xF
22
),2
1,1,
2
1(
12xF
4
)3)(2)(1(!4
)3)(2)(1()3()2)(1(
3
)2)(1(!3
)3)(2)(1()2)(1(2
)1(!2
)1()1(
!11
.0 )(!
)()(),,,(12
x
xxx
rx
r rr
rrxF
![Page 33: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/33.jpg)
2016مارس 15السنة العاشرة العدد مجلة الساتل
93
,,
4321
4
)32
1)(2
2
1)(1
2
1)(
2
1)(24(
)4)(3)(2)(1)(32
1)(2
2
1)(1
2
1)(
2
1(
3
)22
1)(1
2
1)(
2
1)(6(
)3)(2)(1)(22
1)(1
2
1)(
2
1(
2
)12
1)(
2
1)(2(
)2)(1)(12
1)(
2
1(
)2
1(
)1)(2
1(
1
xxxx
x
xxx
23
rr)1(
)1(
)2(...)(
)(
)1(
)(
)1)()...(2)(1()1(
)1(...)(
)(
)1(...)2)(1(1)(
rr
rrr
r
rr
rr)1(
)1(
![Page 34: اـماـج ةـلاد لوـح - al-satil.misuratau.edu.lyal-satil.misuratau.edu.ly/wp-content/uploads/2016/09/0041.pdf · 2016 سرام 15 ددعلا ةرشاعلا ةنسلا](https://reader030.fdocuments.us/reader030/viewer/2022041222/5e0b9ac7db110902df1310bd/html5/thumbnails/34.jpg)
حـــول دالـــة جـامــا
94
املراجع1- Bell W.W Special Functions For Scientist and Engiueers Nostr and
Company,1968.
2- Farrell and Ross, Solved Problems:Gamma and Beta Functions, 1963,New
Yark.
3- George,.E, Richard.A and Ranjan.R . Special Functions, 2004.
4- Snedon I.N Special Functions Of Mathematical Physics,London
Longman,1961.
5- Reichel A.M Special Functions Sedney Science Press,1966.
6- Richard Reals, Roderick Wong, Special Functions, A Graduare Text,
2010,Yale University.
7
2003
8 2006