Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
[ "article:topic", "Bessel functions", "authorname:nwalet", "license:ccbyncsa", "showtoc:no" ]
Mathematics LibreTexts

10.4: Bessel Functions of General Order

  • Page ID
    8330
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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).