Три лекции по теории функций Бесселя. Балакин А.Б. - 22 стр.

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ν
F
ν
(x) ≡
π
2
Z
0
dϕ cos(x sin ϕ)(cos ϕ)
2ν
.
cos(x sin ϕ) F
ν
(x)
F
ν
(x) =
∞
X
n=0
(−1)
n
x
2n
(2n)!
F
(n)
ν
,
F
(n)
ν
≡
π
2
Z
0
dϕ(sin ϕ)
2n
(cos ϕ)
2ν
n ν
z = sin
2
ϕ
F
(n)
ν
=
1
2
1
Z
0
dz z
n−
1
2
(1−z)
ν−
1
2
=
1
2
B
n+
1
2
, ν+
1
2
!
=
Γ(n+
1
2
)Γ(ν+
1
2
)
2Γ(n+ν+1)
.
Γ
n+
1
2
Γ (2n+1)
=
Γ
n+
1
2
(2n)!
=
n−
1
2
· · ·
1
2
· Γ
1
2
(2n)!
=
=
(2n−1) · · · 3 · 1
2
n
(2n)!
·
2n · 2(n − 1) · · · 2 · 1
2
n
n!
√
π =
√
π
2
2n
n!
.
F
ν
(x)
F
ν
(x) =
√
π
2
x
2
!
−ν
Γ
ν+
1
2
!
·
∞
X
n=0
(−1)
n
x
2
2n+ν
Γ(n+1)Γ(n+ν+1)
,
2.3.2. Ïðåäñòàâëåíèå ôóíêöèé Áåññåëÿ ïðîèçâîëüíîãî èíäåêñà ν ñ
                                   ïîìîùüþ èíòåãðàëà Ïóàññîíà

     Äëÿ òîãî, ÷òîáû ïîëó÷èòü èíòåãðàëüíóþ ôîðìóëó Ïóàññîíà äëÿ ôóíê-
öèé Áåññåëÿ ïåðâîãî ðîäà ïðîèçâîëüíîãî èíäåêñà, ðàññìîòðèì èíòåãðàë ñëå-
äóþùåãî âèäà
                                                  π
                                                 Z2
                                   Fν (x) ≡              dϕ cos(x sin ϕ)(cos ϕ)2ν .                   (97)
                                                 0
Ðàñêëàäûâàÿ cos(x sin ϕ) â ñòåïåííîé ðÿä, ïðåäñòàâèì ôóíêöèþ Fν (x) â âèäå
ñóììû
                                                             ∞               x2n (n)
                                                                 (−1)n                                (98)
                                                             X
                                       Fν (x) =                                  F ,
                                                             n=0            (2n)! ν
ãäå âñïîìîãàòåëüíûé èíòåãðàë
                                                         π
                                                      Z2
                                      Fν(n) ≡                dϕ(sin ϕ)2n (cos ϕ)2ν                    (99)
                                                      0

çàâèñèò òîëüêî îò íîìåðà n è èíäåêñà ν . Çàìåíîé ïåðåìåííîé èíòåãðèðîâàíèÿ
z = sin2 ϕ èíòåãðàë (99) ñâîäèòñÿ ê áåòà-ôóíêöèè è âûðàæàåòñÿ ÷åðåç ãàììà-
ôóíêöèè Ýéëåðà:

           1   Z1                                     1    1    1  Γ(n+ 12 )Γ(ν+ 21 )
                                                                                  !
                           n− 12         ν− 12
   Fν(n) =          dz z           (1−z)             = B n+ , ν+ =                    .              (100)
           20                                         2    2    2   2Γ(n+ν+1)

Âñïîìèíàÿ óïðàæíåíèå ñ ãàììà-ôóíêöèåé ïîëóöåëîãî àðãóìåíòà, âûïîëíåí-
íîå â ðàçäåëå 2.1.3. (ôîðìóëû (73)-(75)), ïðèâåäåì îòíîøåíèå ãàììà-ôóíêöèé
ñ óäâîåííûì àðãóìåíòîì ê ñëåäóþùåìó óäîáíîìó âèäó

                         Γ n+ 12
                                    
                                    Γ n+ 12
                                                                      
                                              n− 12 · · · 12 · Γ
                                                                                       
                                                                                        1
                                                                                        2
                                  =         =                                                =
                         Γ (2n+1)    (2n)!          (2n)!
                                                                     √
                                                                √
                                                           
           (2n−1)  · · · 3 · 1     2n · 2(n −  1) · · · 2 · 1          π
        =                     ·                             π =        .                         (101)
              2n (2n)!                      2n n!                   22n n!
Òîãäà ôóíêöèÿ Fν (x) ìîæåò áûòü çàïèñàíà êàê
                     √                                                       2n+ν
                                                                              x
                                                                                        
                       π        x −ν                  1          ∞
                                 !                        !
     Fν (x) =                            Γ ν+           ·      X
                                                                  (−1)n      2         
                                                                                                ,   (102)
                      2 2                             2         n=0      Γ(n+1)Γ(n+ν+1)



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