
1524 SPECIAL FUNCTIONS AND THEIR PROPERTIES
◮ Integrals with Bessel functions.
Let F (a, b, c; x) denote the hypergeometric series (see Section 30.10.1). Then
Z
x
0
x
λ
J
ν
(x) dx =
x
λ+ν+1
2
ν
(λ + ν + 1) Γ(ν + 1)
F
λ + ν + 1
2
,
λ + ν + 3
2
, ν + 1; −
x
2
4
,
where Re(λ + ν) > −1, and
Z
x
0
x
λ
Y
ν
(x) dx = −
cos(νπ)Γ(−ν)
2
ν
π(λ + ν + 1)
x
λ+ν+1
F
λ + ν + 1
2
, ν + 1,
λ + ν + 3
2
; −
x
2
4
−
2
ν
Γ(ν)
λ −ν + 1
x
λ−ν+1
F
λ − ν + 1
2
, 1 − ν,
λ −ν + 3
2
; −
x
2
4
,
where Re λ > |Re ν| − 1.
30.6.3 Zeros and Orthogonality Properties of Bessel Functions
◮ Zeros of Bessel fu nctions.
Each of the functions J
ν
(x) and Y
ν
(x) has infinitely many real zeros (for real ν). All zeros
are simple, except possibly for the