The asymptotic form of the spherical Bessel function $j_{n}(x)$ when $n \to \infty$

asymptoticsdistribution-theoryspecial functions

I am trying to find the asymptotics of the spherical Bessel function $j_n(x)$
when $n\to \infty$.

I was able to find something like

$$ j_{n} ( x) \sim \sqrt{\frac{\pi}{2n}} \delta \left(x – n\right), ~~n\to \infty, $$

where $\delta(x)$ is the Dirac delta function.

My questions are as follows,

  1. Is this true? And if yes, under what condition does this relation
    hold true and how can one derive it?
  2. If it is not true, how can one obtain the correct asymptotic for large values of $n$?

Best Answer

It is $$ j_n (x) = \sqrt {\frac{\pi }{{2x}}} J_{n + \frac{1}{2}} (x) \sim \frac{1}{{\sqrt {2x(2n + 1)} }}\left( {\frac{{ex}}{{2n + 1}}} \right)^{n + \frac{1}{2}} \sim \frac{1}{{2\sqrt 2 n}}\left( {\frac{{ex}}{{2n}}} \right)^n, $$ as $n\to +\infty$ (cf. http://dlmf.nist.gov/10.19.E1). You can obtain this by noting that for large $n$, the Bessel function is controlled by the leading term of its Taylor series and using Stirling's formula for the gamma function to simplify the result.

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