Bessel Functions| Bessel Functions |
In General > s.a. integration [Poisson integral].
$ Def: Solutions of the ordinary
differential equation (Bessel's equation) x2
F''(x) + x F'(x) + (x2
− n2) F(x) = 0.
* Types: First kind,
Jn
and J−n;
Second kind, Yn
and Y−n,
or Nn
and N−n,
a.k.a. Neumann or Weber functions; Third kind,
H 1,2n,
a.k.a. Hankel functions.
* Asymptotic behavior:
Near x = 0, Jn
∝ xn is regular,
Nn
∝ x−n,
or ln x if n = 0, blows up; For x →
∞, Jn and
Nn are oscillatory and go to 0.
* Zeroes: They all have an infinite
number; For Jn(x),
the higher roots are given by xn,k
≈ kπ + (n−\(1\over2\)) π/2.
* Power series expansion:
Jn(x) = (2n n!)−1 xn {1 − [22 1! (n+1)]−1 x2 + [24 2! (n+1) (n+2)]−1 x4 − ...} .
* Recursion relations: They all satisfy
Fn−1(x) + Fn+1(x) = (2n/x) Fn(x) ; dFn(x)/dx = −Fn+1 + (n/x) Fn = \(1\over2\)[Fn−1(x) − Fn+1(x)] .
$ Parseval's integral:
\(J_0(z) = (1/\pi) \int_0^\infty{\rm d}\theta\,\cos(z\cos\theta)\).
@ General references: Watson 44;
in Abramowitz & Stegun 65;
in Arfken 85;
Howls & Daalhuis PRS(99) [asymptotics];
Bailey et al JPA(08)-a0801 [results on moments, and mathematical physics];
Yuste & Abad JPA(11)-a1101 [polynomial approximations].
@ Relationships and related topics:
Mekhfi IJTP(00);
Mekhfi mp/00 [deformed derivatives];
Durand JMP(03)mp/02 [fractional operators];
Cosmin a0912
[integral involving the product of four Bessel functions];
Babusci a1110 [integrals];
Dominici et al PRS(12) [identity involving integrals and sums];
Babusci et al JMP(13)-a1209 [evaluation of sum rules];
Dattoli et al a1311
[products of Bessel functions and their integrals];
> s.a. Whittaker Functions.
Other Related Bessel Functions > s.a. Struve Functions.
* Spherical:
j0(x) = (sin x)/x , j1(x) = (sin x)/x2 − (cos x)/x , j2(x) = 3 (sin x)/x3 − 3 (cos x)/x2 − (sin x)/x .
@ Spherical: Ludu & O'Connell PS(02)mp/01 [Laplace transform];
Boersma & Glasser JPA(05) [differentiation formula];
Mehrem & Hohenegger JPA(10)-a1006 [infinite integral over three spherical Bessel functions];
Mehrem a1110
[integral involving two spherical Bessel functions].
@ Modified: Bender et al JMP(03) [Taylor expansions of powers].
@ Modified, McDonald functions Ka:
Maslanka mp/01 [series representations and fractional derivatives].
@ In Minkowski space: Gerlach PRD(88)gq/99.
@ Other generalizations:
Boyer JMP(69) [Riccati-Bessel functions, zeros];
Lizzi et al JHEP(05)ht [on the fuzzy disk];
Korsch et al JPA(06)qp [2D].
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