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 
k
+
(n–
) /2.
* Power series expansion:
* Recursion relations: They all satisfy
Fn–1(x)
+ Fn+1(x) = (2n/x)
Fn(x) ; dFn(x)/dx
= –Fn+1 + (n/x) Fn
= [Fn–1(x) – Fn+1(x)]
.
$ Parseval's integral: J0(z)
= (1/) 
0infty d
cos(z cos).
@ General references: Watson 66; in Arfken; in Jahnke & Hemde; in
Abramowitz & Stegun;
Howls & Daalhuis PRS(99)
[asymptotics]; Bailey et al JPA(08)-a0801 [results
on moments, and mathematical physics].
@ Relationships:
Mekhfi IJTP(00); > s.a. Whittaker Functions.
@ Related topics: Mekhfi mp/00 [deformed
derivatives]; Durand JMP(03)mp/02 [fractional
operators].
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].
@ 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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1 dec 2008