I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it?


How can I get with Series the asymptotic form for the large real arguments? According to wiki (http://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms) it must be:

$\quad \quad J_n(z)=\sqrt{\frac{2}{\pi z}} \left(e^{\left| \Im(z)\right| } O\left(\frac{1}{\left| z\right| }\right)+\cos \left(\frac{\pi n}{2}-z-\frac{\pi }{4}\right)\right)$

Moreover, though this equation is true, a better approximations may be available for complex $z$. For example, $J_0(z)$, when z is near the negative real line, is approximated better by

$\quad \quad J_0(z) \sim{\sqrt{\frac{-2}{\pi z }} \cos \left(z+\frac{\pi }{4}\right)}$ than by $J_0(z) \sim{\sqrt{\frac{2}{\pi z }} \cos \left(z-\frac{\pi }{4}\right)}$.

Instead of something like that, for the input

Series[BesselJ[0, x], {x, 100 + 3 I, 1}]

I get output of the form

SeriesData[x, Complex[100, 3], 
  {BesselJ[0, Complex[100, 3]], -BesselJ[1,Complex[100, 3]]}, 0, 2, 1]

What am I doing wrong?


closed as off-topic by J. M. will be back soon Mar 31 '17 at 15:03

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – J. M. will be back soon
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  • 2
    $\begingroup$ Have you seen Series[]? $\endgroup$ – J. M. will be back soon May 26 '15 at 16:20
  • 2
    $\begingroup$ Since this is one of the examples in the docs for Series under Scope, I would say this question can be closed. $\endgroup$ – Jens May 26 '15 at 16:47
  • $\begingroup$ @J. M. after your comment I found it under scope, thank you. So, Jens is absolutly right, question can be closed. $\endgroup$ – Hedin May 26 '15 at 17:01
  • $\begingroup$ Probably, I jumped to conclusions. One more sub-question arose. $\endgroup$ – Hedin May 26 '15 at 18:46
  • $\begingroup$ You might want to try expanding with respect to a DirectedInfinity[]. $\endgroup$ – J. M. will be back soon May 27 '15 at 5:22

Your equation for the asymptotic form has a sign error. The correct result is produced by Mathematica:

Normal@Series[BesselJ[n, z], {z, Infinity, 0}]

$$\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{z}} \cos \left(\frac{\pi n}{2}-z+\frac{\pi }{4}\right)$$

A version of this with additional corrections is found in the documentation, as I said in the comments. The other result for $z$ near the negative real line is actually just a special case of this, after you correct the sign error. Alternatively, you can do asymptotic expansions in other infinite directions like this:

Normal@Series[BesselJ[n, z], {z, DirectedInfinity[1 + I], 0}]

$$i \sqrt{\frac{2}{\pi }} e^{i \pi n} \sqrt{\frac{1}{z}} \cos \left(\frac{\pi n}{2}+z+\frac{\pi }{4}\right)$$


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