10.4: Bessel Functions of General Order
- Page ID
- 8330
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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,
\[a_{m} = -\frac{1}{m(m\pm 2\nu)} a_{m-2} \nonumber \]
has the solutions (\(x > 0\))
\[\begin{aligned} J_{\nu}(x) &= \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{\nu+2k}, \\ J_{-\nu}(x) &= \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!\Gamma(-\nu+k+1)} \left(\frac{x}{2}\right)^{-\nu+2k}.\end{aligned} \nonumber \]
The general solution to Bessel’s equation of order \(\nu\) is thus
\[y(x) = A J_{\nu}(x)+BJ_{-\nu}(x), \nonumber \]
for any non-integer value of \(\nu\). This also holds for half-integer values (no logs).