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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
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
λν+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
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Publisher Resources

ISBN: 9781466581494