# Finding limits of derivative spherical bessel function

The derivative of the spherical bessel function is defined as

$f_{n}^{\prime}(z)= - f_{n+1}(z) +(n/z)f_{n}(z).$

The problem occurcs if I try to plot it at z = 0. I want to approximate it using l'Hospital. How do I do this without having a loop of constantly calling the derivative of the spherical bessel function? By using l'Hospital do I need to consider both terms or can I just evaluate the derivative of the second one?

L'Hospital is useful for limits, not plotting. You get a circular reasoning because you can't use L'Hospital.

Here are pictures of the two spherical bessel functions: You can see that there are multiple values in zero.

Then, for the second kind $y$, it goes to $-\infty$: • Sure, but I need to plot it. And in the equation above $z$ is in the denominator. So I need some sort of approximation to plot the zero. Because calling it at zero causes an error. You have a limit for the usual spherical bessel function here dlmf.nist.gov/10.52. So there must be one for the derivative Sep 6, 2018 at 18:45
• @Maxim Use Taylor expansion? Also it's not always defined in zero, but in the neighborhood of zero. Sep 6, 2018 at 18:47
• This is not very easy or practical. Sep 6, 2018 at 18:50
• @Maxim Why? Taylor expansion give you the behaviour near zero very precisely. With order 2 or 3 of the Taylor expansion, you are really close to the actual function. It is VERY used in physics. It IS very easy to plot. Sep 6, 2018 at 18:53
• Because I need to evaluate the derivatives of the derivative of the spherical bessel function up to a few orders. This may be easy for functions which derivatives are short or you can look up, but not for this one. Sep 6, 2018 at 19:01