- #1
IridescentRain
- 16
- 0
Hello.
I'm not terribly proficient with Bessel functions, but I know that those of the first kind are given by
<br /> \begin{eqnarray}<br /> J_n(x) & = & \left(\frac{x}{2}\right)^n\,\sum_{\ell=0}^\infty\frac{(-1)^\ell}{\ell!\,\Gamma(n+\ell+1)}\,\left(\frac{x}{2}\right)^{2\ell},<br /> \end{eqnarray}<br />where \Gamma is the gamma function.
I found in a book that for large x the following is true:
<br /> \begin{eqnarray}<br /> J_n(x) & = & \sqrt{\frac{2}{\pi x}}\,\cos\left(x-\frac{n\pi}{2}-\frac{\pi}{4}\right)\,\left[1+\mathcal{O}(x^{-1})\right]\\<br /> & \approx & \sqrt{\frac{2}{\pi x}}\,\cos\left[x-(2n+1)\,\frac{\pi}{4}\right].<br /> \end{eqnarray}<br />The second line of that last equation is obvious (they just removed every term except the first one from the first line because 1/x^\ell is negligible for large x and \ell\ge1), but where does the cosine come from? How would I begin to prove this?
I'm working with n\in\mathbb{N}\cup\{0\} only, so perhaps this expression is only true for Bessel functions of natural (or zero) order; I wouldn't know. All I know is that for such values of n the gamma function becomes \Gamma(n+\ell+1)=(n+\ell)!.
Thanks in advance for any help!
I'm not terribly proficient with Bessel functions, but I know that those of the first kind are given by
<br /> \begin{eqnarray}<br /> J_n(x) & = & \left(\frac{x}{2}\right)^n\,\sum_{\ell=0}^\infty\frac{(-1)^\ell}{\ell!\,\Gamma(n+\ell+1)}\,\left(\frac{x}{2}\right)^{2\ell},<br /> \end{eqnarray}<br />where \Gamma is the gamma function.
I found in a book that for large x the following is true:
<br /> \begin{eqnarray}<br /> J_n(x) & = & \sqrt{\frac{2}{\pi x}}\,\cos\left(x-\frac{n\pi}{2}-\frac{\pi}{4}\right)\,\left[1+\mathcal{O}(x^{-1})\right]\\<br /> & \approx & \sqrt{\frac{2}{\pi x}}\,\cos\left[x-(2n+1)\,\frac{\pi}{4}\right].<br /> \end{eqnarray}<br />The second line of that last equation is obvious (they just removed every term except the first one from the first line because 1/x^\ell is negligible for large x and \ell\ge1), but where does the cosine come from? How would I begin to prove this?
I'm working with n\in\mathbb{N}\cup\{0\} only, so perhaps this expression is only true for Bessel functions of natural (or zero) order; I wouldn't know. All I know is that for such values of n the gamma function becomes \Gamma(n+\ell+1)=(n+\ell)!.
Thanks in advance for any help!