γλώσσες
Σελίδες
Νομικός
1
1:
BESSEL
1.1. BESSEL
BESSEL
1
2
:
essel (
) Bessel
1 2 , , .
[65] :
[65] :
(1.1) :
.
2
1.2.
BESSEL
Bessel :
:
.
, , (1.5) (1.1)
(1.6) (1.4).
1.3.
BESSEL
BESSEL
1
2
Bessel,
, :
(1.5) , .
(1.7) ,
Bessel 1 2
, , .
(1.7) :
.
3
Bessel 1
[65] :
:
,
,
, Pochhammer.
Bessel 1
Bessel 1 :
:
Bessel 2 (
MacDonald) , [65] :
:
1.4.
Bessel 1 2
[65]:
4
.
5
1.5.
BESSEL
, : ,
, , , ,
, , /,
[18,19,22,25,34,43,53,54,55,59,64].
,
Rosenthal (1962) [54] Ross (1970) [55]
.
,
Penfold (2006) [49].
Q- Marcum,
radar [10,16,17].
,
,
(1984),
.
Gronwall (1932) [26]
. ,
[57]. ,
.
,
.
,
.
6
,
Riccatti.
,
Turn
Bessel.
Turn
Bessel
.
1 -
(DDEs) Bessel
,
Riccati
.
, .
, -
.
, . ,
Turn
. ,
:
.
1.6.
(
) [2,3,28,44,67]
:
7
:
(Completely
Monotonic)
,
(Strictly Completely Monotonic)
,
(Absolutely Monotonic)
,
(Strictly Absolutely Monotonic)
, .
( )
(Log - Convex), (Convex), :
.
(Convex)
.
:
.
( )
(Log - Concave), (Concave),
:
.
(Concave) , .
8
:
T ()
.
A
,
.
M
.
, :
(Berstein - Widder) [67]:
:
,
.
9
2:
BESSEL 1
2.1.
(2.1) Baricz (2010) [12].
,
(1.9). ,
Paris (1984) [48]
, Robert
(1990) [53] , Joshi Bissu (1991) [32]
. , (2.1)
, .
:
H
,
.
(2.2) Baricz (2010) [12]
Bordelon [20] Ross
(1973) [56] Laforgia (1991) [36]
, ,
(1.9). (2.2)
10
.
:
, .
(2.3) Laforgia (1991) [36]
Soni (1965) [62]:
(2.3) Baricz (2010) [12]
,
Gronwall (1932) [26]:
(2.4) Baricz (2010) [12]
(2.3),
Gronwall [26].
. (2.4)
(2.3).
11
(2.5) Baricz Sun (2010) [17]
.
(2.5)
(2.3), (2.5)
,
(2.3).
(2.6) Baricz (2009) [10]
.
,
(2.6)
(2.3). , (2.6)
(2.5).
-
Bessel 1 .
(2.7) (1990)
[29]
(2.1).
12
H (2.8) Joshi Bissu (1996) [33]
.
(2.8)
(2.1) .
(2.9) Sitnik (1995) [60].
Sitnik (2.9)
(2.1).
:
, (2.7)
(2.9).
(2.10) Baricz (2010) [12]
.
,
(2.10) (2.1).
(2.11) Baricz (2010) [12]
.
, (2.11)
(2.1).
13
:
(2.10) (2.11),
, (2.10)
(2.11).
(2.12) Baricz (2010) [12]
. ,
(2.12) (2.3).
,
, (2.12)
(2.4)
, (2.12)
(2.5). ,
, (2.6)
(2.12).
:
, (2.6) (2.12)
, ,
. (2.12)
, .
14
2.2.
(2.13) Laforgia (1991) [36].
Soni [62]:
Soni (1965) Nasell (1974) [45],
. , (2.13)
Bordelon [20] Ross (1973) [56]
, Paris (1984) [48] Joshi Bissu (1991)
[32]
.
H (2.14) Joshi Bissu (1991) [32]
Nasell (1974) [45]:
Soni [62],
. , (2.14)
(2.13).
(2.15) Baricz (2009) [9]
,
,
15
.
(2.15) (2.13)
.
(2.16) Baricz (2010) [12]
,
.
(2.16)
(2.13).
H (2.17) Joshi Bissu (1996) [33]
.
(2.17)
(2.13), .
(2.18) Sitnik (1995) [60].
16
(2.19) Baricz (2010) [12]
.
:
(2.15) (2.19)
.
(2.20) Baricz (2010) [12]
.
, (2.20)
(2.13).
:
(2.19) (2.20),
,
, (2.20)
(2.19).
17
2.3.
H (2.21) Soni (1965) [62]
Nasell (1974) [45]
. ,
Gupta (1966) [27], De Sitter
Goovaerts (1972) [61] , Lorch (1967)
[41] Whittaker. ,
(2.21) Cochran (1967) [21]
Jones [31] Reudink (1968) [52]. , Jones
(1968) :
Cochran (1967) Reudink (1968) :
:
Joshi/Bissu Laforgia (2.21)
Soni (2.13)
, .
(2.22) Nasell (1974) [45]
(2.21) Soni [62].
18
(2.23)
(1990) [29] Mittag-Leffler
Bessel 1 [65]:
Bessel 1
[65]:
(2.24) (1991)
[30] Baricz (2010) [12]
.
, (2.24)
(2.21) Soni.
(2.25) Baricz (2010) [12]
Bessel 1 . ,
, :
19
, (2.25)
(2.24).
:
(2.24) (2.25)
,
, .
(2.26) (2012) [35]
(1.15.iii) (1.15.iv)
(
).
Bessel 1
.
(2.27) (2012) [35]
( ) :
,
,
(2.27)
(2.24),
, .
20
(2.28) (2012) [35]
( )
[65]:
,
:
(2.28) (2.24),
.
(2.29) (2012) [35]
(
).
:
(2.29) (2.28),
.
21
,
Bessel 1 .
(2.30) (2012) [35]
.
(2.31) Segura (2011) [57].
,
:
,
:
:
(2.31) :
22
.
(2.32) Segura (2011) [57]
:
, (2.31).
(2.32)
.
2.4.
Bessel 1
.
(2.33)
(1990) [29] Mittag-Leffler
Bessel 1 [64]:
23
Bessel 1
:
(2.34) Baricz (2010) [12]
, .
(2.35) Baricz (2010) [12]
, . ,
(2012) [35]
( ,
).
(2.36) (2012) [35]
(1.15.iii) (1.15.iv),
(
), Mittag-Leffler
Bessel 1 [65]:
24
[65]:
:
Bessel 1
. (2.34)
Laforgia Natalini (2010) [40] Segura (2011) [57].
(2.37) (2012) [35],
(2.27)
(
) [65]:
,
,
(2.37)
(2.34),
, .
(2.38) (2012) [35]
, (2.28),
25
(
) [65]:
,
:
(2.38) (2.34),
.
(2.39) (2012) [35]
, (2.29),
(
).
:
(2.39) (2.38),
.
,
Bessel 1 .
(2.40) (2012) [35]
. ,
26
,
. [65]:
:
:
(2012) [35] ,
:
, .
,
, ,
. ,
:
(2.41) (2.42) Segura (2011)
[57]
(2.31).
:
27
,
.
:
(2.42)
(2.41), :
,
.
2.5.
(2.43) (2012) [35]
(
), (1.15.iv),
Bessel (1.7). ,
(2.43) Segura (2011) [57]
(2.31).
28
(2.44) Segura (2011) [57]
:
(2.31).
(2.44)
. :
, (2.44) Gronwall
(1932) [26], . , Phillips
Malin (1950) [51] , . ,
(2.44) Amos (1974) [4],
.
:
(1.15.iii) (1.15.iv),
(2.44)
Turn:
(2.45) Segura (2011) [57]
:
29
(2.31).
(2.45)
. :
(2.45) Amos [4],
.
2.6.
(2.46) Gronwall (1932) [26].
:
(2.46) Gronwall Laforgia
(2.4),
.
:
(2.5)
Gronwall,
.
30
(2.47) Baricz Sun (2009) [16]
Gronwall [26] Nasell [45]
(2.6).
:
(2.46) (2.47)
,
(2.47) .
(2.48) (2012) [35]
(1.15.iv)
,
.
(2.48)
, , .
(2.49) (2.50) Segura (2011)
[57] :
(2.31).
31
,
, .
(2.49) (2012)
[35]
(
), (1.15.iv),
Bessel (1.7).
Baricz (2009) [9] Laforgia/Natalini (2010) [40]
.
:
(2.50)
,
.
(2.51) Segura (2011) [57]
:
Turn:
Bessel 1 .
(2.52) Phillips Malin (1950) [51]
Turn:
32
,
Thiruvenkatachar Nanjundiah (1951), Amos (1974) [4],
Joshi/Bissu (1991) [32] Baricz (2010)
[12].
:
(2.49)
(2.52)
(2.53) Segura (2011) [57]
:
(2.31).
(2.53) ,
:
.
33
2.7.
(2.54) Segura (2011) [57]
(1.15.iii) (1.15.iv)
Riccatti:
(2.54).
:
, :
(2.54),
.
34
2.8.
(2.55) Laforgia (1991) [36]
(2.3) (2.13).
(2.56) Joshi Bissu (1996) [33]
(1.15.iv)
(2.55)
.
(2.3).
(2.57) Baricz (2010) [12]
(2.4) (2.19). T
(2.57)
(2.55),
.
35
:
A
, (2.56) (2.57)
.
(2.58) Baricz (2010) [12]
(2.4) (2.20).
, (2.58) (2.57).
:
A
, (2.56) (2.58)
.
(2.59) Baricz (2010) [12]
(2.6) (2.15)
. , (2.59)
(2.55).
36
,
(2.59)
Laforgia (2.56).
:
A
, (2.56) (2.59)
.
2.9.
(2.60) Laforgia (1991) [36]
(2.3) (2.13).
(2.61) Joshi Bissu (1996) [33]
(1.15.iv)
.
37
:
O Joshi Bissu
,
(2.61)
(2.60). .
.
. ,
,
. ,
(2.61) .
(2.62) Baricz (2010) [12]
(2.4) (2.19).
, (2.62)
(2.60).
(2.63) Baricz (2010) [12]
(2.4) (2.20)
(2.62).
38
(2.64) Baricz (2010) [12]
(2.15) (2.19).
(2.60). , ,
. , (2.64)
(2.62).
2.10. O TURAN
(2.65) Baricz (2009) [6]
( ) .
.
(2.65) Thiruvenkatachar Nanjundiah
(1951) [63], Amos (1974) [4] Joshi/Bissu (1991)
[32]. , Segura (2011) [57]
(2.31).
:
, Lorch (1994) [42] (2.65)
Turn (generalized
Turnian):
Turn (2.65) .
39
(2.66) ,
Thiruvenkatachar Nanjundiah (1951) [63], Amos (1974) [4],
Joshi Bissu (1991) [32] Baricz
(2010) [12] Segura (2011) [57].
:
,
Turn (2.66) .
(2.67) Baricz (2010) [12]
, .
(2.68) Baricz (2010) [12]
, .
Segura (2011) [57] ,
, (2.31)
40
Bessel 1 .
. (2.65)
(2.66)
.
41
3:
BESSEL 2
3.1.
(3.1) Laforgia (1991) [36]
.
Ross (1970) [55],
Bordelon (1973) [20] Paris (1984) [48].
:
,
,
.
H (3.2) Laforgia (1991) [36]
o
, ,
. , (3.2)
(3.1).
42
(3.3) Joshi Bissu (1991) [32]
(2.1)
,
.
(3.3)
(3.1).
:
H
, ,
.
(3.4) Joshi Bissu (1991) [32]
,
.
(3.5) Laforgia (1991) [36]
,
(3.1) (3.2). (3.1)
.
,
,
43
. , ,
(3.5)
.
:
,
,
,
.
(3.6) Laforgia (1991) [36]
Soni
Bessel 2 :
.
(3.7) Joshi Bissu (1996) [33].
, (3.5)
(3.7).
:
Bessel 2 :
44
(3.8) Baricz (2010) [12]
,
. ,
Weniger Cizek (1989) [66]
[24]:
.
,
(3.8) (3.1),
.
H (3.9) (3.8).
(3.10) Miller Samko (2001) [44]
.
(3.10)
Laforgia (3.5),
. ,
, (3.10)
(3.5) (2.5)
Bessel 1 .
,
(3.10)
(3.8).
45
:
:
.
3.2.
(3.11) ,
Bessel 1 Soni (1965)
[62].
(3.12) Segura (2011) [57]
.
,
, :
.
46
(3.13) - (3.16) Segura (2011) [57]
:
:
(3.13) (3.14)
, ,
(3.15) (3.16)
, .
(3.13) ,
.
(3.14)
,
(3.15)
.
47
:
, ,
,
.
:
,
.
(3.17) Segura (2011) [57]
Bessel 2 .
:
(3.16)
(3.17).
48
3.3.
(3.18) Segura (2011) [57],
(1.16.iii) (1.16.iv)
Riccatti:
(3.18).
(3.18).
(3.19) (3.18).
49
3.4.
(3.20) Segura (2011) [57]
:
(3.13) (3.16).
(3.22)
.
(3.21) Segura (2011) [57]
:
(3.13) (3.16).
(3.21)
.
50
:
(3.20)
(3.21),
(3.20) .
(3.22) Segura (2011) [57]
:
(3.13) (3.16).
(3.22)
.
(3.23) Segura (2011) [57]
:
(3.13) (3.16).
(3.23)
.
51
(3.24) Segura (2011) [57]
Bessel 2
.
:
(3.23)
(3.24).
3.5. O TURAN
Segura (2011) [57]
, ,
Bessel 1
, .
(3.26) Segura (2011) [57]
,
:
52
,
,
, (3.26) .
:
Turn :
.
E Turn
. , .
:
,
.
(3.27) Segura (2011) [57]
:
. :
,
53
:
, (3.27)
.
(3.28) Segura (2011) [57]. ,
. :
,
. ,
, :
(3.29) Segura (2011) [57]
(3.28),
54
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