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) + (x2n2) 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, Jnxn is regular, Nnx−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,kkπ + (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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