Bessel Functions |

**In General** > s.a. integration [Poisson integral].

$ __Def__: Solutions of the
ordinary differential equation (Bessel's equation) *x*^{2} *F''*(*x*)
+ *x* *F'*(*x*) + (*x*^{2}–*n*^{2}) *F*(*x*)
= 0.

* __Types__: First kind,
*J*_{n} and *J*_{–n};
Second kind, *Y*_{n} and *Y*_{–n},
or *N*_{n} and *N*_{–n},
a.k.a. Neumann or Weber functions; Third kind, *H*^{ 1,2}_{n},
a.k.a. Hankel functions.

* __Asymptotic behavior__: Near *x* =
0, *J*_{n} ∝ *x*^{n}
is regular, *N*_{n} ∝ *x*^{–n},
or ln *x* if *n* = 0, blows up;
For *x* → ∞, *J*_{n}
and *N*_{n} are oscillatory and go to 0.

* __Zeroes__: They all have
an infinite number; For *J*_{n}(*x*),
the higher roots are given by *x*_{n,k }≈ *k*π +
(*n*–\(1\over2\)) π/2.

* __Power series expansion__:

*J*_{n}(*x*) = (2^{n} *n*!)^{–1} *x*^{n} {1 – [2^{2}_{} 1! (*n*+1)]^{–1} *x*^{2} + [2^{4} 2! (*n*+1) (*n*+2)]^{–1} *x*^{4} – ...} .

* __Recursion relations__: They all satisfy

*F*_{n–1}(*x*)
+ *F*_{n+1}(*x*) = (2*n*/*x*)
*F*_{n}(*x*) ; d*F*_{n}(*x*)/d*x*
= –*F*_{n+1} + (*n*/*x*) *F*_{n}
= \(1\over2\)[*F*_{n–1}(*x*) – *F*_{n+1}(*x*)]
.

$ __Parseval's integral__: *J*_{0}(*z*)
= (1/π) ∫_{0}^{∞} d*θ* cos(*z* cos*θ*).

@ __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__:

*j*_{0}(*x*) =
(sin *x*)/*x* , *j*_{1}(*x*)
= (sin *x*)/*x*^{2} – (cos *x*)/*x* , *j*_{2}(*x*)
= 3 (sin *x*)/*x*^{3} – 3 (cos *x*)/*x*^{2} – (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 K__

@

@

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