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10.4 Bessel functions of general order

The recurrence relation for the Bessel function of general order ± ν can now be solved by using the gamma function,

{a}_{m} = − {1\over m(m ± 2ν)}{a}_{m−2}
(10.27)

has the solutions (x > 0)

\begin{eqnarray} {J}_{ν}(x)& =& {\mathop{∑ }}_{k=0}^{∞} {{(−1)}^{k}\over k!Γ(ν + k + 1)}{\left ({x\over 2}\right )}^{ν+2k}, %&(10.28) \\ {J}_{−ν}(x)& =& {\mathop{∑ }}_{k=0}^{∞} {{(−1)}^{k}\over k!Γ(−ν + k + 1)}{\left ({x\over 2}\right )}^{−ν+2k}.%&(10.29) \\ \end{eqnarray}

The general solution to Bessel’s equation of order ν is thus

y(x) = A{J}_{ν}(x) + B{J}_{−ν}(x),
(10.30)

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

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