Уравнения математической физики. Сборник задач. Даишев Р.А - 57 стр.

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P
m
n
(cos θ) n
(0, π) sin θ :
π
Z
0
P
n
i
(cos θ)P
n
k
(cos θ) sin θdθ = 0, (i 6= k);
π
Z
0
[P
n
k
(cos θ)]
2
sin θdθ =
2(k + n)!
(2k + 1)(k − n)!
.
(a) J
ν
(x) =
∞
X
k=0
(−1)
k
³
x
2
´
ν+2k
Γ(k + 1)Γ (k + ν + 1)
;
(b) N
ν
(x) =
J
ν
(x) cos πν − J
−ν
(x)
sin πν
.
J
−n
(x) = (−1)
n
J
n
(x).
J
1
2
(x) .
J
−
1
2
(x) .
d
dx
h
J
ν
(x)
x
ν
i
= −
J
ν+1
(x)
x
ν
.
J
0
0
(x) = −J
1
(x).
d
dx
[x
ν
J
ν
(x)] = x
ν
J
ν−1
(x).
d
dx
[xJ
1
(x)] = xJ
0
(x).
J
3
2
(x) .
J
5
2
(x) .
N
−
1
2
(x) .
J
0
(0), J
n
(0), J
0
0
(0), J
0
n
(0).
   Ôóíêöèè Pnm (cos θ) ïðè ôèêñèðîâàííîì n îðòîãîíàëüíû äðóã
äðóãó íà èíòåðâàëå (0, π) ñ âåñîì sin θ :
             Zπ
                  Pin (cos θ)Pkn (cos θ) sin θdθ = 0, (i 6= k);
             0

             Zπ
                                                       2(k + n)!
                  [Pkn (cos θ)]2 sin θdθ =                          .
                                                   (2k + 1)(k − n)!
             0




   81. Îïðåäåëèòü îáëàñòè ñõîäèìîñòè ðÿäîâ:
                                                      ³ ´ν+2k
                                 ∞                     x
                                 X                     2
                                          k
            (a) Jν (x) =               (−1)                             ;
                                 k=0          Γ(k + 1)Γ (k + ν + 1)

                            Jν (x) cos πν − J−ν (x)
                   (b) Nν (x) =                     .
                                     sin πν
   82. Äîêàçàòü: J−n (x) = (−1)n Jn (x).
   83. Âû÷èñëèòü J 1 (x) .
                   2
   84. Âû÷èñëèòü J− 1 (x) .
                            h2     i
                      Jν (x)
   85.   Äîêàçàòü:     d
                      dxxν
                              = − Jν+1 xν
                                         (x)
                                             .
   86.                J00
         Äîêàçàòü: (x) = −J1 (x).
                       d
   87.   Äîêàçàòü: [xν Jν (x)] = xν Jν−1 (x).
                      dx
                       d
   88.   Äîêàçàòü: [xJ1 (x)] = xJ0 (x).
                      dx
   89.   Âû÷èñëèòü: J 3 (x) .
                      2
   90.   Âû÷èñëèòü: J 5 (x) .
                      2
   91.   Âû÷èñëèòü : N− 1 (x) .
                          2
   92.   Âû÷èñëèòü J0 (0), Jn (0), J00 (0), Jn0 (0).
   93.   Èñïîëüçóÿ ðåêóðåíòíûå ñîîòíîøåíèÿ, âûðàçèòü:

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