My Math Forum Roots of a transcendental equation

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November 20th, 2013, 06:31 AM   #1
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Roots of a transcendental equation

Hello!

I have an equation $J_1(a/x)*Y_0(b/x)-J_0(b/x)*Y_1(a/x)=0$ and need to find its roots. Here $J_\alpha(x)$ and $Y_\alpha(x)$ are Bessel functions (well, a Bessel and Neumann functions, respectively). It's obvious that the equation has infinitely many roots, and the plot of this function (in case of $a=10, b=1$) is below the text of this message. Ho do I get the roots? I need a fast and accurate method. Basic numerical methods don't work here.

What have I tried? Well, I use Mathematica with its nice command "FindRoot", bet there are some difficulties with its use in this case (especially when x is small). Second, the series of these functions are too complex to analyse them. And third, as I said, standard numerical methods (for example, Newton's method) can't help here. I haven't met this equation in any book/textbook.

Since $J_\alpha(\infty)=Y_\alpha(\infty)=0,$ it makes sence to get the roots from right to left and stop the process at a certain point.

I've found a quite similar equation here (the last one in the list). One can obtain a lot of roots in a couple of seconds using this calculator. The good news for me is that Mathematica is used there. How do they do that?
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