Методы математической физики - 144 стр.

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N
ν
(x) =
J
ν
(x) cos νπ − J
−ν
(x)
sin νπ
,
ν
0/0 ν → n
N
n
(x) = lim
ν→n
J
ν
(x) cos νπ − J
−ν
(x)
sin νπ
=
= lim
ν→n
∂
∂ν
[J
ν
(x) cos νπ − J
−ν
(x)]
∂
∂ν
[sin νπ]
=
=
1
π
·
∂
∂ν
J
ν
(x)
¯
¯
¯
¯
ν=n
− (−1)
n
∂
∂ν
J
−ν
(x)
¯
¯
¯
¯
ν=n
¸
.
∂
∂ν
J
ν
(x)
¯
¯
¯
¯
ν=n
=
∞
X
m=0
(−1)
m
Γ(m + 1)
³
x
2
´
2m
×
×
∂
∂ν
·
³
x
2
´
ν
1
Γ(m + ν + 1)
¸
¯
¯
¯
¯
ν=n
= J
n
(x) ln
x
2
− R
n
(x) ,
∂
∂ν
J
−ν
(x)
¯
¯
¯
¯
ν=n
= −J
−n
(x) ln
x
2
+ R
−n
(x) ,
R
n
(x) =
∞
X
m=0
(−1)
m
Γ
0
(m + n + 1)
Γ(m + 1) [Γ(m + n + 1)]
2
³
x
2
´
2m+n
.
J
−n
(x) = (−1)
n
J
n
(x)
N
n
(x) =
1
π
h
2 J
n
(x) ln
x
2
− R
n
(x) − (−1)
n
R
−n
(x)
i
.

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