Editing Bessel function (section)

=== Spherical Hankel functions: ''h''{{su|b=''n''|p=(1)}}, ''h''{{su|b=''n''|p=(2)}} <span class="anchor" id="Spherical Hankel functions"></span> ===
[[File:Plot of the spherical Hankel function of the first kind h n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Hankel function of the first kind {{math|''h''{{su|b=''n''|p=(1)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
[[File:Plot of the spherical Hankel function of the second kind h n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the spherical Hankel function of the second kind {{math|''h''{{su|b=''n''|p=(2)}}(''x'')}} with {{math|1=''n'' = −0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]

There are also spherical analogues of the [[#Hankel functions|Hankel functions]]:
<math display="block">\begin{align}
h_n^{(1)}(x) &= j_n(x) + i y_n(x), \\
h_n^{(2)}(x) &= j_n(x) - i y_n(x).
\end{align}</math>

There are simple closed-form expressions for the Bessel functions of [[half-integer]] order in terms of the standard [[trigonometric function]]s, and therefore for the spherical Bessel functions. In particular, for non-negative integers {{mvar|n}}:
<math display="block">h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!\,(2x)^m} \frac{(n+m)!}{(n-m)!},</math>
and {{math|''h''{{su|b=''n''|p=(2)}}}} is the complex-conjugate of this (for real {{mvar|x}}). It follows, for example, that {{math|1=''j''<sub>0</sub>(''x'') = {{sfrac|sin ''x''|''x''}}}} and {{math|1=''y''<sub>0</sub>(''x'') = −{{sfrac|cos ''x''|''x''}}}}, and so on.

The spherical Hankel functions appear in problems involving [[spherical wave]] propagation, for example in the [[electromagnetic wave equation#Multipole expansion|multipole expansion of the electromagnetic field]].