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Ploting zero order Bessel function

  1. Sep 30, 2004 #1
    Hello guys, i had a little chat with a teacher of mine and he asked me how can someone plot the zero order Bessel function. Here is what i've done..

    using the integral expresion for [tex]J_{0}(r)[/tex]
    [tex]J_{0}(r)=\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)d\theta[/tex]
    i can calculate the first order derivative with respect to r
    [tex]\frac {\partial}{\partial r}J_{0}(r)=-\frac {1}{\pi}\int_0^\pi \sin(r\cos\theta)\cos\theta d\theta[/tex]
    wich when evaluated in r=0 is 0. For the second derivative
    [tex]\frac {\partial^2}{\partial r^2}J_{0}(r)=-\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)\cos^{2}\theta d\theta[/tex]
    wich evaluated in r=0 is equal to -1/2.
    The idea is to construct the taylor series around r=0. And given the fact that
    i can easily bound the error, ie, if i only take two terms of the series
    [tex]|E(r)|\leq \frac{r^4}{4!\pi}[/tex]
    so, if, for instance, i want to know where is the first zero of the function, given the first approximation, i can say that is on 2 with an error of 0.21....
    given the next term
    [tex]|E(r)|\leq \frac{r^6}{6!\pi}[/tex]
    tells me that the zero is in 2^(3/2) with an error of 0.23
    and so on...

    do you guys think this is a correct procedure?

    is there any other way i can construct the plot?

    i really want to impress my teacher, so any help would be well received.

    Last edited: Sep 30, 2004
  2. jcsd
  3. Oct 1, 2004 #2


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    Have you thought about creating a table of Bessel functions?
  4. Oct 1, 2004 #3
    Looks good, you have derived the small series representation for Jo :

    \mbox{\Huge \[
    J_\nu (z) = \left( {\frac{z}{2}} \right)^\nu \sum\limits_{k = 0}^\infty {\frac{{\left( { - z^2 /4} \right)^k }}{{k!\Gamma (\nu + k + 1)}}}

    Unfortunately, this will only converge up to about 10 on the real axis with 10 signifigant digit calculator. For arguments larger an asymptotic form must be used : Hankel's Simiconvergent Asymptotic Expansion is probably the best.

  5. Oct 1, 2004 #4
    Heh... i know that. That was exactly the point my teacher wanted to make.

    He said to me: "ok, you know how it behaves for small values of r and for large values, but what hapens with regular values of r? how can you see the graph of the function?"

    But yeah, i think ill need more terms to get a nice aproximation of the first zero *at least*.

    Do u guys know any other method that i might consider?

    Thx for the help :)