While learning about Bessel functions I've came up with the following argument:

Since the Bessel equation is $$y''+\frac{1}{x} y'+ \left(1-\frac{\alpha^2}{x^2} \right) y=0, $$ one might expect that in the limit $x \to \infty$ we have $y''+y \approx 0$, which is the equation modelling simple harmonic motion (aka linear combinations of sines and cosines) of period $2 \pi$. Thus, we should have $y \approx C_1 \cos x+ C_2 \sin x$ for large $x$.

I'm aware that there is a branch of mathematics, called asymptotics (?) that deals with such ideas. Nonetheless I have several questions:

  1. How can you know what terms are to be omitted as $x$ gets large? perhaps $y'$ grows faster than $\frac{1}{x}$ decays (so that the middle term cannot be omitted).

  2. Looking at the graph of the Bessel function $J_0(x)$ one can see that the amplitude of the oscillation decays. Is there a way to predict that, and know the rate at which $C_1,C_2$ decay?

  3. In general, is there a rigorous statement of the form: If $y' \approx G(x,y)$ is an approximation of the differential equation $y'=F(x,y)$ at the point $x_0$, with a solution $y=\varphi$, then the original equation has a solution $y$ which is $\approx \varphi$ near $x_0$? Hopefully with a precise notion of what "$\approx$" means.

Thank you!

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    $\begingroup$ The answer to these questions are quite involved and I am not sure it can be reasonable answered in the context of this forum. I can only point you to the book "Advanced mathematical methods for scientists and engineers" by Bender and Orszag where the questions are answered (part of it in Sec 3.7, Ex 3). $\endgroup$ – Fabian Jul 8 '15 at 9:56

I do not know how much this will help you; so, please, forgive me if I am off-topic.

The solution of the differential equation $$y''+\frac{1}{x} y'+ \left(1-\frac{\alpha^2}{x^2} \right) y=0$$ is given by $$y=c_1 J_{\alpha }(x)+c_2 Y_{\alpha }(x)$$ where appears the Bessel functions of the first and second kinds.

Considering the case where $x$ is large, the following first order approximations are available $$J_{\alpha }(x)\approx \sqrt{\frac{2}{\pi x }} \cos \left(\frac{\pi \alpha }{2}+\frac{\pi }{4}-x\right)$$ $$Y_{\alpha }(x)\approx -\sqrt{\frac{2}{\pi x }} \sin \left(\frac{\pi a}{2}+\frac{\pi }{4}-x\right)$$

Have a look here and here for more details about Hankel's expansions.

  • $\begingroup$ Thanks Claude, this helps with some of my questions. $\endgroup$ – user1337 Jul 9 '15 at 4:43

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