Ploting zero order Bessel function

[ReyChiquito]

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 $$J_{0}(r)$$
$$J_{0}(r)=\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)d\theta$$
i can calculate the first order derivative with respect to r
$$\frac {\partial}{\partial r}J_{0}(r)=-\frac {1}{\pi}\int_0^\pi \sin(r\cos\theta)\cos\theta d\theta$$
which when evaluated in r=0 is 0. For the second derivative
$$\frac {\partial^2}{\partial r^2}J_{0}(r)=-\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)\cos^{2}\theta d\theta$$
which 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
$$|J^{(n)}(r)|\leq\frac{1}{\pi}$$
i can easily bound the error, ie, if i only take two terms of the series
$$J_{0}(r)=1-\frac{r^2}{4}+E(r^4)$$
where
$$|E(r)|\leq \frac{r^4}{4!\pi}$$
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
$$J_{0}(r)=1-\frac{r^2}{4}+\frac{r^4}{64}-E(r^6)$$
where
$$|E(r)|\leq \frac{r^6}{6!\pi}$$
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.

Thx.

 

[Tide]

Have you thought about creating a table of Bessel functions?

 

[flexten]

Looks good, you have derived the small series representation for Jo :

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

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

Best

 

[ReyChiquito]

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 :)