Три лекции по теории функций Бесселя. Балакин А.Б. - 15 стр.

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x
k
x
−1
d
dx
!
k
[x
ν
J
ν
(x)] = x
ν−k
J
ν−k
(x) ,
x
−1
d
dx
!
k
h
x
−ν
J
ν
(x)
i
= (−1)
k
x
−ν−k
J
ν+k
(x) .
Z
dx x
ν
J
ν−1
(x) = x
ν
J
ν
(x) ,
Z
dx x
−ν
J
ν+1
(x) = −x
−ν
J
ν
(x) .
Y
ν
(x)
J
ν
(x)
sin πν = −sin π(ν −1) cos πν = −cos π(ν −1)
J
ν
Y
ν
I
ν
(x)
d
dx
[x
ν
I
ν
(x)] = x
ν
I
ν−1
(x) ,
d
dx
h
x
−ν
I
ν
(x)
i
= x
−ν
I
ν+1
(x) ,
2I
0
ν
(x) = I
ν−1
(x)+I
ν+1
(x) , I
ν−1
(x)−I
ν+1
(x) =
2ν
x
I
ν
(x) ,
x
−1
d
dx
!
k
[x
ν
I
ν
(x)] = x
ν−k
I
ν−k
(x) ,
x
−1
d
dx
!
k
h
x
−ν
I
ν
(x)
i
= x
−ν−k
I
ν+k
(x) ,
Z
dx x
ν
I
ν−1
(x) = x
ν
I
ν
(x) ,
Z
dx x
−ν
I
ν+1
(x) = x
−ν
I
ν
(x) .
     Èç äèôôåðåíöèàëüíûõ ñîîòíîøåíèé äëÿ ôóíêöèé Áåññåëÿ (56) âûòåêà-
þò äâà âàæíûõ ñëåäñòâèÿ. Âî-ïåðâûõ, ðàçäåëèì ñîîòíîøåíèÿ (56) íà x, ïðî-
äèôôåðåíöèðóåì èõ åùå ðàç, çàòåì ïîâòîðèì óêàçàííóþ îïåðàöèþ íóæíîå
÷èñëî ðàç. Òîãäà î÷åâèäíûìè ñòàíîâÿòñÿ äèôôåðåíöèàëüíûå ñîîòíîøåíèÿ
k -ãî ïîðÿäêà
                                            !k
                              −1        d
                              x                  [xν Jν (x)] = xν−k Jν−k (x) ,
                                       dx
                     d k h −ν
                                  !
                         −1
                          x Jν (x) = (−1)k x−ν−k Jν+k (x) .        (59)
                                  i
                 x
                    dx
Âî-âòîðûõ, èíòåãðèðóÿ ñîîòíîøåíèÿ äëÿ ôóíêöèé Áåññåëÿ (56), ïîëó÷èì èç-
âåñòíûå íåîïðåäåëåííûå èíòåãðàëû
         Z                                              Z
             dx xν Jν−1 (x) = xν Jν (x) ,                       dx x−ν Jν+1 (x) = −x−ν Jν (x) .   (60)

Ôóíêöèè Áåññåëÿ âòîðîãî ðîäà Yν (x) ïîä÷èíÿþòñÿ òåì æå áàçîâûì äèôôå-
ðåíöèàëüíûì ñîîòíîøåíèÿì (56), ÷òî è ôóíêöèè Jν (x). Äëÿ òîãî, ÷òîáû â
ýòîì óáåäèòüñÿ, äîñòàòî÷íî âçÿòü îïðåäåëåíèå (34), âîñïîëüçîâàòüñÿ ôîðìó-
ëàìè (56) ñ ó÷åòîì òîãî, ÷òî sin πν = − sin π(ν − 1) è cos πν = − cos π(ν − 1).
Ýòî îçíà÷àåò, ÷òî âñå îñòàëüíûå ñîîòíîøåíèÿ (57)-(60) òàêæå íå ìåíÿþò ñâî-
åãî âèäà ïðè çàìåíå Jν íà Yν .
     Ðåêóððåíòíûå ñîîòíîøåíèÿ äëÿ Iν (x), - ôóíêöèé Áåññåëÿ ìíèìîãî àð-
ãóìåíòà, ïîëó÷àþòñÿ àíàëîãè÷íî, íî òåïåðü îíè îñíîâûâàþòñÿ íà äèôôå-
ðåíöèðîâàíèè ôîðìóëû (43). Ëåãêî óáåäèòüñÿ, ÷òî ïîÿâëÿþùèåñÿ îòëè÷èÿ
ñâÿçàíû ñ èçìåíåíèåì çíàêà âî âòîðîì ðàâåíñòâå (56):
              d ν                                            d h −ν
                [x Iν (x)] = xν Iν−1 (x) ,                      x Iν (x) = x−ν Iν+1 (x) ,
                                                                        i

             dx                                             dx
                                                                                    2ν
             2Iν0 (x) = Iν−1 (x)+Iν+1 (x) ,                 Iν−1 (x)−Iν+1 (x) =        Iν (x) ,
                                                                                    x
                                            !k
                                  −1    d
                              x                  [xν Iν (x)] = xν−k Iν−k (x) ,
                                       dx
                                        !k
                                d
                           x−1                   x−ν Iν (x) = x−ν−k Iν+k (x) ,
                                             h              i

                               dx
             Z                                              Z
                 dx xν Iν−1 (x) = xν Iν (x) ,                   dx x−ν Iν+1 (x) = x−ν Iν (x) .    (61)



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