Editing Bessel function (section)

=== Riccati–Bessel functions: ''S<sub>n</sub>'', ''C<sub>n</sub>'', ''ξ<sub>n</sub>'', ''ζ<sub>n</sub>'' <span class="anchor" id="Riccati–Bessel functions"></span> ===
[[Jacopo Riccati|Riccati]]–Bessel functions only slightly differ from spherical Bessel functions:
<math display="block">\begin{align}
S_n(x)     &= x j_n(x) = \sqrt{\frac{\pi x}{2}} J_{n+\frac{1}{2}}(x) \\
C_n(x)     &= -x y_n(x) = -\sqrt{\frac{\pi x}{2}} Y_{n+\frac{1}{2}}(x) \\
\xi_n(x)   &= x h_n^{(1)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(1)}(x) = S_n(x) - iC_n(x) \\
\zeta_n(x) &= x h_n^{(2)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(2)}(x) = S_n(x) + iC_n(x)
\end{align}</math>
[[File:Riccati Bessel Function S 3D Complex Color Plot with Mathematica 13.2.svg|alt=Riccati–Bessel functions  Sn complex plot from -2-2i to 2+2i|thumb|Riccati–Bessel functions Sn complex plot from −2&nbsp;−&nbsp;2''i'' to 2&nbsp;+&nbsp;2''i'']]
They satisfy the differential equation
<math display="block">x^2 \frac{d^2 y}{dx^2} + \left (x^2 - n(n + 1)\right) y = 0.</math>

For example, this kind of differential equation appears in [[quantum mechanics]] while solving the radial component of the [[Schrödinger equation]] with hypothetical cylindrical infinite potential barrier.<ref>Griffiths. Introduction to Quantum Mechanics, 2nd edition, p.&nbsp;154.</ref> This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as [[Mie scattering]] after the first published solution by Mie (1908). See e.g., Du (2004)<ref>{{cite journal |first=Hong |last=Du |title=Mie-scattering calculation |journal=Applied Optics |volume=43 |issue=9 |pages=1951–1956 |date=2004 |doi=10.1364/ao.43.001951 |pmid=15065726 |bibcode=2004ApOpt..43.1951D}}</ref> for recent developments and references.

Following [[Peter Debye|Debye]] (1909), the notation {{mvar|ψ<sub>n</sub>}}, {{mvar|χ<sub>n</sub>}} is sometimes used instead of {{mvar|S<sub>n</sub>}}, {{mvar|C<sub>n</sub>}}.