Три лекции по теории функций Бесселя. Балакин А.Б. - 16 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

sin πν=−sin π(ν−1)
K
ν
(x)
d
dx
[x
ν
K
ν
(x)] = −x
ν
K
ν−1
(x) ,
d
dx
h
x
−ν
K
ν
(x)
i
= −x
−ν
K
ν+1
(x) ,
−2K
0
ν
(x) = K
ν−1
(x)+K
ν+1
(x) , K
ν+1
(x)−K
ν−1
(x) =
2ν
x
K
ν
(x) ,
x
−1
d
dx
!
k
[x
ν
K
ν
(x)] = (−1)
k
x
ν−k
K
ν−k
(x) ,
x
−1
d
dx
!
k
h
x
−ν
K
ν
(x)
i
= (−1)
k
x
−ν−k
K
ν+k
(x) ,
Z
dx x
ν
K
ν−1
(x) = −x
ν
K
ν
(x) ,
Z
dx x
−ν
K
ν+1
(x) = −x
−ν
K
ν
(x) .
ν = n
J
n
(x) J
n−1
(x)
J
n−2
(x) J
n−1
(x) J
n−2
(x) J
n−3
(x)
J
n
(x)
J
0
(x) J
1
(x)
J
n
(x) = P
n−2
1
x
!
J
0
(x) + Q
n−1
1
x
!
J
1
(x) ,
P
n−2
1
x
Q
n−1
1
x
n
J
2
(x) = −J
0
(x) +
2
x
J
1
(x) ,
J
3
(x) = −
4
x
J
0
(x) +
−1 +
8
x
2
!
J
1
(x) , ...
J
−n
J
n
Èç (61) ñ ó÷åòîì îïðåäåëåíèÿ (44) è ðàâåíñòâà sin πν=− sin π(ν−1) ïîëó÷àåì
àíàëîãè÷íûå ñîîòíîøåíèÿ äëÿ ôóíêöèé Ìàêäîíàëüäà Kν (x):
       d ν                                              d h −ν
         [x Kν (x)] = −xν Kν−1 (x) ,                       x Kν (x) = −x−ν Kν+1 (x) ,          (62)
                                                                   i

      dx                                               dx
                                                                                 2ν
     −2Kν0 (x) = Kν−1 (x)+Kν+1 (x) ,                       Kν+1 (x)−Kν−1 (x) =      Kν (x) ,   (63)
                                                                                 x
                                      !k
                            −1    d
                        x                  [xν Kν (x)] = (−1)k xν−k Kν−k (x) ,                 (64)
                                 dx
                                  !k
                     −1      d
                                       x−ν Kν (x) = (−1)k x−ν−k Kν+k (x) ,                     (65)
                                     h                 i
                    x
                            dx
     Z                                                 Z
         dx xν Kν−1 (x) = −xν Kν (x) ,                      dx x−ν Kν+1 (x) = −x−ν Kν (x) .    (66)

Ôîðìóëû (61)-(66) ðåêîìåíäóåòñÿ ïðîâåðèòü ñàìîñòîÿòåëüíî.

     2.1.2. Ïðèëîæåíèå ðåêóððåíòíûõ ñîîòíîøåíèé ê ôóíêöèÿì
                            Áåññåëÿ öåëîãî èíäåêñà (ν = n)

      Èñïîëüçóÿ ðåêóððåíòíûå                       ñîîòíîøåíèÿ, ïîëó÷åííûå â           ïðåäûäóùåì
ðàçäåëå, ëþáóþ ôóíêöèþ Jn (x) öåëîãî èíäåêñà ìîæíî âûðàçèòü ÷åðåç Jn−1 (x)
è Jn−2 (x).  ñâîþ î÷åðåäü Jn−1 (x) âûðàæàåòñÿ ÷åðåç Jn−2 (x) è Jn−3 (x) è òàê
äàëåå. Â ðåçóëüòàòå ôóíêöèÿ Áåññåëÿ Jn (x) ìîæåò áûòü ïðåäñòàâëåíà êàê
ëèíåéíàÿ êîìáèíàöèÿ äâóõ ôóíêöèé J0 (x) è J1 (x):
                                                  1               1
                                                   !                     !
                    Jn (x) = Pn−2                   J0 (x) + Qn−1   J1 (x) ,                   (67)
                                                  x               x
â êîòîðîé ñèìâîëàìè          Pn−2 x1          è                 îáîçíà÷åíû ïîëèíîìû ñîîòâåòñòâó-
                                  
                                                  Qn−1 x1
                                                       


þùåé ñòåïåíè îò îáðàòíîé âåëè÷èíû àðãóìåíòà. Ñòðóêòóðà ýòèõ ïîëèíîìîâ
ñòàíîâèòñÿ ïîíÿòíîé, åñëè ïðèâåñòè íåñêîëüêî ïðèìåðîâ äëÿ ìàëûõ çíà÷åíèé
n:
                                           2
                         J2 (x) = −J0 (x) + J1 (x) ,
                                           x
                             4               8
                                                !
                 J3 (x) = − J0 (x) + −1 + 2 J1 (x) , ...              (68)
                             x               x
Åñëè ìû èìååì äåëî ñ ôóíêöèåé Áåññåëÿ îòðèöàòåëüíîãî èíäåêñà J−n , òî ïî-
ìîùüþ ñîîòíîøåíèÿ (33) îíà ìîæåò áûòü ñâåäåíà ê Jn , à çàòåì ïðåäñòàâëåíà



                                                           16

Страницы