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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?


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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 $\tan x$.

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