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I noticed, numerically, the following relationship between Bessel function of the first kind $J_\nu(z)$ and Bessel function of the second kind $Y_\nu(z)$ for $\nu \in \mathbb{Z}$ and $z > 0$

$$2J_\nu(-z) = \mathrm{Im}[Y_\nu(-z)]$$

For example, for $\nu = 0,1,2$, I plotted the left and right expressions in Mathematica with the following code

For[n = 0, n < 3, n++, 
 Plot[{2*BesselJ[n, -x], Im[BesselY[n, -x]]}, {x, 0, 5}, 
   PlotLegends -> "Expressions"] // Print]

plots

I looked up the properties of Bessel functions on Wikipedia but couldn't see anything that directly demonstrates this relationship. Could anyone help prove it/disprove it? I'm not a mathematician, so apologies if this is a trivial question.

neuro630
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  • Answering my own question if anyone else is also wondering: this relationship can be shown rather easily as a consequence of equations 9.1.35 and 9.1.36 in the Handbook of Mathematical Function by Abramowitz and Stegun (nineth printing), which states $J_\nu(ze^{m\pi i}) = e^{m \nu \pi i} J_\nu(z)$ and $Y_\nu(ze^{m \pi i}) = e^{-m \nu \pi i}Y_\nu(z) + 2i \sin(m \nu \pi) \cot(\nu \pi)J_\nu(z)$

    for $m \in \mathbb{Z}$. (This was originally my answer to the question, but was deleted by moderators.)

     – neuro630 Mar 19 '23 at 05:39

0 Answers0