Три лекции по теории функций Бесселя. Балакин А.Б. - 10 стр.

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J
n
J
−n
ν
2
=
n
2
Y
ν
(x)
Y
ν
(x) ≡
cos πνJ
ν
(x) − J
−ν
(x)
sin πν
.
Y
ν
(x)
J
ν
(x) Y
ν
(x)
W [J
ν
(x), Y
ν
(x)] = −
1
sin πν
W [J
ν
(x), J
−ν
(x)] =
2
πx
,
ν
y(x) = C
1
J
ν
(x) + C
2
Y
ν
(x) .
Y
n
(x)
Y
n
= lim
ν→n
Y
ν
cos πν → (−1)
n
sin πν → 0
0
0
Y
n
(x) =
1
π
lim
ν→n
"
−π tan πν J
ν
(x) +
∂
∂ν
J
ν
(x) −
1
cos πν
∂
∂ν
J
−ν
(x)
#
,
Y
n
(x) =
2
π
J
n
(x) log
x
2
!
−
1
π
n−1
X
m=0
x
2
!
2m−n
(n−m−1)!
m!
−
−
1
π
∞
X
m=0
(−1)
m
x
2
2m+n
m!(m+n)!
Γ
0
(m+1)
Γ(m+1)
+
Γ
0
(n+m+1)
Γ(n+m+1)
.
1.2.2. Ôóíêöèè Áåññåëÿ âòîðîãî ðîäà - ôóíêöèè Âåáåðà-Øëåôëè

       Ôóíêöèè Áåññåëÿ öåëîãî èíäåêñà Jn è J−n ëèíåéíî çàâèñèìû è ïîòîìó
íå îáðàçóþò ôóíäàìåíòàëüíîé ñèñòåìû ðåøåíèé óðàâíåíèÿ Áåññåëÿ ñ ν 2 =
n2 . Äëÿ òîãî, ÷òîáû îáîéòè ýòó ïðîáëåìó, áûëè ââåäåíû òàê íàçûâàåìûå
ôóíêöèè Áåññåëÿ âòîðîãî ðîäà Yν (x) êàê ëèíåéíûå êîìáèíàöèè ñëåäóþùåãî
âèäà
                               cos πνJν (x) − J−ν (x)
                      Yν (x) ≡                        .              (34)
                                      sin πν
Î÷åâèäíî, ÷òî â ñèëó ëèíåéíîñòè óðàâíåíèÿ Áåññåëÿ ôóíêöèÿ Yν (x), êàê ëè-
íåéíàÿ êîìáèíàöèÿ ðåøåíèé, òàêæå ÿâëÿåòñÿ åãî ðåøåíèåì. Ýòè ôóíêöèè
ïðèíÿòî íàçûâàòü èìåíàìè Âåáåðà è Øëåôëè. Òåðìèí ôóíêöèè Íåéìàíà,
ââåäåííûé äëÿ ýòèõ ôóíêöèé, íàïðèìåð, â ó÷åáíèêå [4], ïî-âèäèìîìó, íåäî-
ñòàòî÷íî îáîñíîâàí ñ èñòîðè÷åñêîé òî÷êè çðåíèÿ [1,5]. Äåòåðìèíàíò Âðîí-
ñêîãî, ïîäñ÷èòàííûé äëÿ ïàðû ôóíêöèé Jν (x) è Yν (x):
                                   1                          2
              W [Jν (x), Yν (x)] = −   W [Jν (x), J−ν (x)] =    ,     (35)
                                sin πν                       πx
íå îáðàùàåòñÿ â íóëü íè ïðè êàêèõ çíà÷åíèÿõ èíäåêñîâ. Ïîýòîìó îáùåå ðåøå-
íèå óðàâíåíèÿ Áåññåëÿ (1) äëÿ ëþáîãî çíà÷åíèÿ èíäåêñà ν ñòàíäàðòíî ïðåä-
ñòàâëÿåòñÿ â âèäå
                          y(x) = C1 Jν (x) + C2 Yν (x) .                    (36)
Äëÿ òîãî, ÷òîáû ÿâíî ïðåäñòàâèòü ðàçëîæåíèå ôóíêöèé Yn (x), îáû÷íî ïîëü-
çóþòñÿ ïðåäåëîì Yn = ν→n
                     lim Yν . Â ýòîì ïðåäåëå cos πν → (−1)n , sin πν → 0,
ñëåäîâàòåëüíî, ñ ó÷åòîì ñîîòíîøåíèÿ (33) ïîëó÷àåì íåîïðåäåëåííîñòü òèïà
0.   Âîñïîëüçîâàâøèñü ïðàâèëîì Ëîïèòàëÿ (L'Hospital)
0


             1                        ∂             1    ∂
                    "                                                   #
               lim −π tan πν Jν (x) +
     Yn (x) = ν→n                        Jν (x) −           J−ν (x) ,       (37)
             π                        ∂ν          cos πν ∂ν
èñêîìóþ ôóíêöèþ ïðèâîäÿò ê ñëåäóþùåìó ñòàíäàðòíîìó âèäó
                                            n−1       !2m−n
                    2           x   1             x           (n−m−1)!
                                   !
                                            X
            Yn (x) = Jn (x) log   −                                    −
                    π           2   π       m=0   2              m!
                             2m+n
                             x
                                    Γ0 (m+1) Γ0 (n+m+1) 
                                                                   
               1 X∞
             −      (−1)  m  2             +             .       (38)
               π m=0     m!(m+n)! Γ(m+1) Γ(n+m+1)
Çäåñü è äàëåå øòðèõ ñèìâîëèçèðóåò ïðîèçâîäíóþ îò óêàçàííîé ôóíêöèè ïî
å¼ àðãóìåíòó. Âûâîä ýòîé ôîðìóëû íå âõîäèò â îáÿçàòåëüíóþ ÷àñòü íàøåé

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