Page 780 - Handbook Of Integral Equations

Wronskians and similar formulas

2 2
W(J ν , J –ν )= – sin(πν), W(J ν , Y ν )= ,
πx πx
2 sin(πν) 2
J ν (x)J –ν+1 (x)+ J –ν (x)J ν–1 (x)= , J ν (x)Y ν+1 (x) – J ν+1 (x)Y ν (x)= – .
πx πx
Here the notation W(f, g)= fg – f g is used.

x x
Integral representations
The functions J ν and Y ν can be represented in the form of definite integrals (for x > 0):
π ∞
πJ ν (x)= cos(x sin θ – νθ) dθ – sin πν exp(–x sinh t – νt) dt,
0 0
π
∞
πY ν (x)= sin(x sin θ – νθ) dθ – (e νt + e –νt cos πν)e –x sinh t dt.
0 0
1
For |ν| < , x >0,
2
x
2 1+ν –ν ∞ sin(xt) dt
J ν (x)= 1 2 ν+1/2 ,
π 1/2 Γ( – ν) 1 (t – 1)
2
x
2 1+ν –ν ∞ cos(xt) dt
Y ν (x)= – 1 2 ν+1/2 .
π 1/2 Γ( – ν) 1 (t – 1)
2
1
For ν > – ,
2
2(x/2) ν π/2 2ν
J ν (x)= cos(x cos t) sin tdt (Poisson’s formula).
1
π 1/2 Γ( + ν) 0
2
For ν =0, x >0,
2 ∞ 2 ∞
J 0 (x)= sin(x cosh t) dt, Y 0 (x)= – cos(x cosh t) dt.
π 0 π 0
For integer ν = n =0, 1, 2, ... ,
1 π
J n (x)= cos(nt – x sin t) dt (Bessel’s formula),
π 0
π/2
2

J 2n (x)= cos(x sin t) cos(2nt) dt,
π
0
2 π/2
J 2n+1 (x)= sin(x sin t) sin[(2n +1)t] dt.
π 0
Integrals with Bessel functions
x λ+ν+1 2
λ x λ + ν +1 λ + ν +3 x
x J ν (x) dx= ν F , , ν+1; – , Re(λ+ν)>–1,
0 2 (λ + ν +1) Γ(ν +1) 2 2 4
where F(a, b, c; x) is the hypergeometric series (see Section 10.9 of this supplement),
x cos(νπ)Γ(–ν) λ + ν +1 λ + ν +3 x 2
λ
x Y ν (x) dx = – ν x λ+ν+1 F , ν +1, , –
0 2 π(λ + ν +1) 2 2 4
ν
2 Γ(ν) λ–ν+1 λ – ν +1 λ – ν +3 x 2
– x F ,1 – ν, , – , Re λ > |Re ν| – 1.
λ – ν +1 2 2 4

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 765