BesselJ0

Explicit plot of BesselY0 (red), BesselJ1 (green) and BesselJ0 (blue)

$u+\mathrm i v=\mathrm{BesselJ}_0 (x+\mathrm i y)$

BesselJ0 called also the Bessel function of zero order and also $J_0$, is entire function, solution of equation

$ \!\!\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~

J_0(z)+ J_0'(z)/z+ J_0(z)=0 $

$ \!\!\!\!\!\!\!\!\!\! (2) \displaystyle ~ ~ ~ J_0(0)=1 ~, ~~ J_0'(0)=0$

Complex map of function BesselJ0 is shown in figure at right. The short notation $J_0=\mathrm{BesselJ0}$ is also valid.

Integral representation

$ \!\!\!\!\!\!\!\!\!\! (3) \displaystyle ~ ~ ~ \mathrm{BesselJ}_0(z)=

\frac{1}{\pi}\int_0^\pi \cos\!\big( z \cos (t) \big)~ \mathrm d t $

Symmetry

The BesselJ0 is symmetric function,

$ \!\!\!\!\!\!\!\!\!\! (4) \displaystyle ~ ~ ~ \mathrm{BesselJ}_0'(0)=1$

In particular,

$ \!\!\!\!\!\!\!\!\!\! (5) \displaystyle ~ ~ ~ \mathrm{BesselJ}_0(-z)= \mathrm{BesselJ}_0(z)$

Expansion at zero

The straightforward Taylor expansion at zero can be written as follows:

$ \!\!\!\!\!\!\!\!\!\! (6) \displaystyle ~ ~ ~ \mathrm{BesselJ}_0(z)=

\sum_{n=0}^{\infty} \frac{ (-z^2/4)^n } {\mathrm{Factorial}(n)^2} = 1 -\frac{z^2}{4} +\frac{z^4}{64} -\frac{z^6}{2304}+ ... $

The series converges in the whole complex plane and, at the complex(double) arithmetics, gives of order of dozen significant figures at least for $|z|\!<\!20$; it is sufficient to keep few tens of terms in the expansion. For larger values of the argument, the expansion below can be used.

Expansion at infinity

Expansion at large vales of the argument can be written as follows:

$ \!\!\!\!\!\!\!\!\!\! (8) \displaystyle ~ ~ ~ J_0(z)=

\sqrt{\frac{2}{\pi z}} \cos\!\left(z-\frac{\pi}{4}\right) \left( 1 - \frac{9}{128 z^2}+ ..\right) + \sqrt{\frac{2}{\pi z}} \sin\!\left(z-\frac{\pi}{4}\right) \frac{1}{8z}\left( 1 - \frac{75}{128 z^2}+ ..\right) $

Mathematica allows to calculate a dozen of term of this expansion; they can be extracted also from the complex(double) implementation Besselj0.cin.

Behavior along the real axis

Along the real axis, BesselJ0 oscillates (like other Bessel functions). The zeros of are denoted with $j_{0,n}$; where $n$ is supposed to be positive integer. The WebMatematica offers the on-line service to evaluate these zeros ; a thousand of zeros can be evaluated within few seconds. In particular, the first 5 values are

2.404825557695773

5.520078110286311

8.653727912911011

11.79153443901428

14.93091770848779

Integral properties

The orthogonality properties for the zero-th Bessel is suggested in the following form:

$ \displaystyle \int_0^W ~ J_0(s_p r)~ j_0(s_q r) ~r~ \mathrm d r = \delta_{p,q} \frac{W^2}{2} J_0'(s_p W)^2$

$ \displaystyle \int_0^W ~ J_0(s r_q)~ j_0(s r_p) ~r~ \mathrm d r = \delta_{p,q} \frac{S^2}{2} J_0'(S\, r_p )^2$

$ r_{2n+1}=W ~ ~ , ~ ~ s_{n+1}=S$

$ \zeta_{2n+1}=SW$

That needs to be somehow interpreted.

Application

The Bessel function appear at the consideration of wave equations with circular symmetry.

Keywords

BesselJ1, BesselY0, BesselH0 Bessel function