Verify . . . . .(complex analysis)

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Complex analysis: Let J_n (z) be the Bessel function for a positive integer n of order n. Verify?

J_n-1 (z) + J_n+1 (z) = ((2n)/z) J_n (z)
 

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Bessel Identity...


For a Bessel function of the first kind [tex]J_n(z)[/tex]

Identity confirmed:
[tex]J_{n-1}(z) + J_{n+1}(z) = \frac{2\,n\,J_{n}(z)}{z} \; \; \; n > 0 \; \; \; z \neq 0[/tex]

[tex]\Mfunction{BesselJ}(-1 + n,z) + \Mfunction{BesselJ}(1 + n,z) = \frac{2\,n\,\Mfunction{BesselJ}(n,z)}{z} \; \; \; n > 0 \; \; \; z \neq 0 [/tex]

n = 1
Attachment 1: LHS plot
Attachment 2: RHS plot

The x-intercepts and amplitudes appear to match, therefore this is an identity.

Reference:
http://www.efunda.com/math/bessel/besselJYPlot.cfm
 

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