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# 10.4: Bessel Functions of General Order

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The recurrence relation for the Bessel function of general order $$\pm\nu$$ can now be solved by using the gamma function,

$a_{m} = -\frac{1}{m(m\pm 2\nu)} a_{m-2}$

has the solutions ($$x > 0$$)

\begin{aligned} J_{\nu}(x) &= \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{\nu+2k}, \\ J_{-\nu}(x) &= \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(-\nu+k+1)} \left(\frac{x}{2}\right)^{-\nu+2k}.\end{aligned}

The general solution to Bessel’s equation of order $$\nu$$ is thus

$y(x) = A J_{\nu}(x)+BJ_{-\nu}(x),$

for any non-integer value of $$\nu$$. This also holds for half-integer values (no logs).