Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm stuck on the following problem. Find the solutions of the equation $$J_0(x)-J_1(x)=0,$$ where $J$ is the Bessel function of the first kind. Is there any method to solve it in closed form or do I have to find the solutions numerically?

Thanks.

share|improve this question

1 Answer 1

up vote 4 down vote accepted

There is no closed expression for the roots of this equation. We can say something about the roots that a numerical study won't tell us explicitly.

Expand in large $x$. We find $\frac{1}{\sqrt{16\pi x^3}}((8x-2)\cos x - \sin x) \sim 0,$ so $$\tan x \sim 8x.$$ The right hand side is large by assumption, so the roots are $$\begin{equation*} x_n\approx \frac{(2n-1)\pi}{2},\tag{1} \end{equation*}$$ for $n\in\mathbb{N}$. (These are the vertical asymptotes of $\tan x$, see the figure below.) In fact, this approximation works well even for small $n$.

Below we give some of the roots to six digits.

$$\begin{array}{ccc} n & x_n & (2n-1)\pi/2 \\ \hline 1 & 1.43470 & 1.57080 \\ 2 & 4.68010 & 4.71239 \\ 4 & 10.9832 & 10.9956 \\ 8 & 23.5564 & 23.5619 \\ 16 & 48.6921 & 48.6947 \\ 32 & 98.9589 & 98.9602 \\ 64 & 199.491 & 199.491 \\ 128 & 400.553 & 400.553 \end{array}$$

enter image description here

Figure 1. Plot of $8x$ and $\tan x$. Notice the curves intersect roughly at the asymptotes of $\tan x$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.