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In writing up a report that includes the modified spherical Bessel functions, the standard notation I've seen is to write them as

\begin{align} i_n(r) &= \sqrt{\frac{\pi}{2r}} I_{n+1/2}(r) \\ k_n(r) &= \sqrt{\frac{\pi}{2r}} K_{n+1/2}(r) \end{align} with the functions $I_\nu(r)$ and $K_\nu(r)$ defined as \begin{align} I_{\nu}(r) &= i^{-\nu} J_{\nu}(ir)\\ K_{\nu}(r) &= \frac{\pi}{2 \sin \nu \pi} (I_{-\nu}(r) - I_{\nu}(r)) \end{align}

and note that the $i$ in $I_v$ is $\sqrt{-1}$ while $i_n$ is a function (and $J_\nu$ is the standard Bessel function). Short of changing the name of the functions, is there a better way to express $i$ and $i_\nu$?

Update: This particular notation comes from the paper (paywall)

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up vote 2 down vote accepted

If you look at MathWorld's definition, I would adopt that notation given they cite a pretty standard reference.

That is, Abramowitz and Stegun. Handbook of Mathematical Functions (see Section 10).

I would certainly recommend using that in your report as as reference, so there is no confusion.

I would also remark that the Gov through NIST is creating the Digital Library of Mathematical Functions and that is a nice reference.

For your example, see: (this may also be a good reference).


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The problem is still there though, they just get around it by not using them at the same time. They still have a function named $i_n$ and at some point you need to define $i=\sqrt{-1}$. Look at eq. (11) on the mathworld link. – Hooked Jan 4 '13 at 19:30
@Hooked: precisely, I would point to the standard and make an explicit note regarding that, so the reader does not confuse it. That way, you are still using the standard and noting that confusing usage. You might even want to italicize the point when you give the definition. For example, in engineering, they sometimes use the letter $j$ instead if $i$, because $i$ = current, and it is somewhat standard in that domain. Regards. – Amzoti Jan 4 '13 at 19:33
It's worth noting that in this case that sometimes $j_v$ is used to define the spherical bessel functions (eq. 11 mathworld again)! I may stick with A&S formula and just cite them. Thanks for the reference, I didn't know it was hosted online (nor that is was commissioned by the US government)! – Hooked Jan 4 '13 at 19:53
@Hooked: Please see my update regarding the Gov angle (no pun intended). Regards! – Amzoti Jan 4 '13 at 21:22
+ 1 I can't wait to hear how all goes with your talk! – amWhy May 9 '13 at 0:33

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