# Wronskian of Bessel Functions of non-integral order v, -v

by mjordan2nd
Tags: bessel, functions, nonintegral, order, wronskian
 P: 118 My textbook states $$J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x}$$ My textbook derives this by showing that $$J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x}$$ where C is a constant. C is then ascertained by taking x to be very small and using only the first order of the power series expansion for Bessel functions. Does this mean that this computation for C is inexact? It seems that there should be some error terms in there from higher powers of x, or am I missing something? By the way, I'm using Arfken/Weber and N.N. Lebedev as my guide here. Thanks for any help. Edit: Perhaps this would have been better in the differential equations section?
 P: 756 Wronskian of Bessel Functions of non-integral order v, -v $$J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x}$$ is not an approximate. It is an identity for any variable x and any order v Considering a constant order v, then $$C=-\frac{2 \sin v \pi}{\pi}$$ is constant. hence $$J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x}$$