Dependencies within Bessel's recursive relationshipby kzhu Tags: bessel, dependencies, recursive, relationship 

#1
Oct3009, 04:54 PM

P: 12

Dear All,
I am working on the wave equation in the spherical coordinate and come across with the recursive relationship of Bessel's function, which are given in four expressions [tex]B_\nu(x) = \frac{x}{2\nu}\left(B_{\nu1}(x)+B_{\nu+1}(x)\right)[/tex] [tex]B_\nu'(x) = \frac{\nu}{x}B_\nu(x)B_{\nu+1}(x) [/tex] [tex]B_\nu'(x) = \frac{\nu}{x}B_\nu(x)+B_{\nu1}(x) [/tex] [tex]B_\nu'(x) = \frac{1}{2}\left(B_{\nu1}B_{\nu+1}(x)\right) [/tex] where [tex]B_\nu (x)[/tex] represents the Bessel/Neumann/Hankel function. I notice that two of these four recursive relationship can be derived from the other two. My guess extends that, given a 2nd order linear PDE similar to Bessel's equation, can we draw the conclusion that there would be 2 linearly independent recursive relationship? Thank you for any feedback. kzhu 


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