Editing Bessel function (section)

=== Spherical Bessel functions: ''j<sub>n</sub>'', ''y<sub>n</sub>'' <span class="anchor" id="Spherical Bessel functions"></span> ===
[[File:Plot of the spherical Bessel function of the first kind j n(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 Bessel function of the first kind {{math|''j<sub>n</sub>''(''z'')}} with {{math|1=''n'' = 0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]
[[File:Plot of the spherical Bessel function of the second kind y n(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 Bessel function of the second kind {{math|''y<sub>n</sub>''(''z'')}} with {{math|1=''n'' = 0.5}} in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} with colors created with Mathematica 13.1 function ComplexPlot3D]]

[[File:Sphericalbesselj.png|thumb|350px|right|Spherical Bessel functions of the first kind <math> j_\alpha(x)</math>, for <math>\alpha = 0,1,2</math>.]]
[[File:Sphericalbessely.png|thumb|350px|right|Spherical Bessel functions of the second kind <math> y_\alpha(x)</math>, for <math>\alpha = 0,1,2</math>.]]

When solving the [[Helmholtz equation]] in spherical coordinates by separation of variables, the radial equation has the form
<math display="block">x^2 \frac{d^2 y}{dx^2} + 2x \frac{d y}{dx} +\left(x^2 - n(n + 1)\right) y = 0.</math>

The two linearly independent solutions to this equation are called the '''spherical Bessel functions''' {{mvar|j<sub>n</sub>}} and {{mvar|y<sub>n</sub>}}, and are related to the ordinary Bessel functions {{mvar|J<sub>n</sub>}} and {{mvar|Y<sub>n</sub>}} by<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_437.htm p. 437, 10.1.1].</ref>
<math display="block">\begin{align}
j_n(x) &= \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x), \\
y_n(x) &= \sqrt{\frac{\pi}{2x}} Y_{n+\frac{1}{2}}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-\frac{1}{2}}(x).
\end{align}</math>

{{mvar|y<sub>n</sub>}} is also denoted {{mvar|n<sub>n</sub>}} or {{mvar|[[Eta (letter)|η]]<sub>n</sub>}}; some authors call these functions the '''spherical Neumann functions'''.

From the relations to the ordinary Bessel functions it is directly seen that:
<math display="block">\begin{align}
j_n(x) &= (-1)^{n} y_{-n-1} (x)
\\
y_n(x) &= (-1)^{n+1} j_{-n-1}(x)
\end{align}</math>

The spherical Bessel functions can also be written as ('''{{va|Rayleigh's formulas}}''')<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.25, 10.1.26].</ref>
<math display="block">\begin{align}
j_n(x) &= (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}, \\
y_n(x) &= -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\cos x}{x}.
\end{align}</math>

The zeroth spherical Bessel function {{math|''j''<sub>0</sub>(''x'')}} is also known as the (unnormalized) [[sinc function]]. The first few spherical Bessel functions are:<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.11].</ref>
<math display="block">\begin{align}
j_0(x) &= \frac{\sin x}{x}. \\
j_1(x) &= \frac{\sin x}{x^2} - \frac{\cos x}{x}, \\
j_2(x) &= \left(\frac{3}{x^2} - 1\right) \frac{\sin x}{x} - \frac{3\cos x}{x^2}, \\
j_3(x) &= \left(\frac{15}{x^3} - \frac{6}{x}\right) \frac{\sin x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\cos x}{x}
\end{align}</math>
and<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.12].</ref>
<math display="block">\begin{align}
y_0(x) &= -j_{-1}(x) = -\frac{\cos x}{x}, \\
y_1(x) &= j_{-2}(x)  = -\frac{\cos x}{x^2} - \frac{\sin x}{x}, \\
y_2(x) &= -j_{-3}(x) = \left(-\frac{3}{x^2} + 1\right) \frac{\cos x}{x} - \frac{3\sin x}{x^2}, \\
y_3(x) &= j_{-4}(x)  = \left(-\frac{15}{x^3} + \frac{6}{x}\right) \frac{\cos x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\sin x}{x}.
\end{align}</math>

The first few non-zero roots of the first few spherical Bessel functions are:
{| class="wikitable sortable"
|+ Non-zero Roots of the Spherical Bessel Function (first kind)
! Order !! Root 1 !! Root 2 !! Root 3 !! Root 4 !! Root 5
|-
| <math>j_{0}</math> || 3.141593 || 6.283185 || 9.424778 || 12.566371 || 15.707963
|-
| <math>j_{1}</math> || 4.493409 || 7.725252 || 10.904122 || 14.066194 || 17.220755
|-
| <math>j_{2}</math> || 5.763459 || 9.095011 || 12.322941 || 15.514603 || 18.689036
|-
| <math>j_{3}</math> || 6.987932 || 10.417119 || 13.698023 || 16.923621 || 20.121806
|-
| <math>j_{4}</math> || 8.182561 || 11.704907 || 15.039665 || 18.301256 || 21.525418
|}

{| class="wikitable sortable"
|+ Non-zero Roots of the Spherical Bessel Function (second kind)
! Order !! Root 1 !! Root 2 !! Root 3 !! Root 4 !! Root 5
|-
| <math>y_{0}</math> || 1.570796 || 4.712389 || 7.853982 || 10.995574 || 14.137167
|-
| <math>y_{1}</math> || 2.798386 || 6.121250 || 9.317866 || 12.486454 || 15.644128
|-
| <math>y_{2}</math> || 3.959528 || 7.451610 || 10.715647 || 13.921686 || 17.103359
|-
| <math>y_{3}</math> || 5.088498 || 8.733710 || 12.067544 || 15.315390 || 18.525210
|-
| <math>y_{4}</math> || 6.197831 || 9.982466 || 13.385287 || 16.676625 || 19.916796
|}

==== Generating function ====
The spherical Bessel functions have the generating functions<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.39].</ref>
<math display="block">\begin{align}
\frac{1}{z} \cos \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), \\
\frac{1}{z} \sin \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} y_{n-1}(z).
\end{align}</math>

==== Finite series expansions ====
In contrast to the whole integer Bessel functions {{math|''J''<sub>n</sub>(''x''), ''Y''<sub>n</sub>(''x'')}}, the spherical Bessel functions {{math|''j''<sub>n</sub>(''x''), ''y''<sub>n</sub>(''x'')}} have a finite series expression:<ref>L.V. Babushkina, M.K. Kerimov, A.I. Nikitin, Algorithms for computing Bessel functions of half-integer order with complex arguments, [https://www.sciencedirect.com/science/article/pii/0041555388900183 p. 110, p. 111].</ref>
<math display="block">\begin{alignat}{2}
j_n(x) &= \sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x) = \\
&= \frac{1}{2x} \left[ e^{ix} \sum_{r=0}^n \frac{i^{r-n-1}(n+r)!}{r!(n-r)!(2x)^r} + e^{-ix} \sum_{r=0}^n \frac{(-i)^{r-n-1}(n+r)!}{r!(n-r)!(2x)^r} \right] \\
&= \frac{1}{x} \left[ \sin\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} + \cos\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \\

y_n(x) &= (-1)^{n+1} j_{-n-1}(x) = (-1)^{n+1} \frac{\pi}{2x}J_{-\left(n+\frac{1}{2}\right)}(x) =  \\
&= \frac{(-1)^{n+1}}{2x} \left[ e^{ix} \sum_{r=0}^n \frac{i^{r+n}(n+r)!}{r!(n-r)!(2x)^r} + e^{-ix} \sum_{r=0}^n \frac{(-i)^{r+n}(n+r)!}{r!(n-r)!(2x)^r} \right] = \\
&= \frac{(-1)^{n+1}}{x} \left[ \cos\left(x+\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} - \sin\left(x+\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right]
\end{alignat}</math>

==== Differential relations ====
In the following, {{mvar|f<sub>n</sub>}} is any of {{mvar|j<sub>n</sub>}}, {{mvar|y<sub>n</sub>}}, {{math|''h''{{su|b=''n''|p=(1)}}}}, {{math|''h''{{su|b=''n''|p=(2)}}}} for {{math|1=''n'' = 0, ±1, ±2, ...}}<ref>Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.23, 10.1.24].</ref>
<math display="block">\begin{align}
\left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{n+1} f_n(z)\right ) &= z^{n-m+1} f_{n-m}(z), \\
\left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{-n} f_n(z)\right ) &= (-1)^m z^{-n-m} f_{n+m}(z).
\end{align}</math>