Editing Bessel function (section)

=== Bessel functions of the second kind: ''Y<sub>α</sub>'' <span class="anchor" id="Weber functions"></span><span class="anchor" id="Neumann functions"></span><span class="anchor" id="Bessel functions of the second kind"></span> ===
[[File:Besselyn.png|thumb|350px|Plot of Bessel function of the second kind, <math>Y_\alpha(x)</math>, for integer orders <math>\alpha = 0, 1, 2</math>]]
The Bessel functions of the second kind, denoted by {{math|''Y<sub>α</sub>''(''x'')}}, occasionally denoted instead by {{math|''N<sub>α</sub>''(''x'')}}, are solutions of the Bessel differential equation that have a singularity at the origin ({{math|1=''x'' = 0}}) and are [[multivalued function|multivalued]]. These are sometimes called '''Weber functions''', as they were introduced by {{harvs|txt|authorlink=Heinrich Martin Weber|first=H. M.|last=Weber|year=1873}}, and also '''Neumann functions''' after [[Carl Neumann]].<ref name="mhtlab.uwaterloo.ca">{{cite web |url=http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-date=2022-10-09 |url-status=live |title=Bessel Functions of the First and Second Kind |website=mhtlab.uwaterloo.ca |access-date=24 May 2022 |page=3}}</ref>

For non-integer {{mvar|α}}, {{math|''Y<sub>α</sub>''(''x'')}} is related to {{math|''J<sub>α</sub>''(''x'')}} by
<math display="block">Y_\alpha(x) = \frac{J_\alpha(x) \cos (\alpha \pi) - J_{-\alpha}(x)}{\sin (\alpha \pi)}.</math>

In the case of integer order {{mvar|n}}, the function is defined by taking the limit as a non-integer {{mvar|α}} tends to {{mvar|n}}:
<math display="block">Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x).</math>

If {{mvar|n}} is a nonnegative integer, we have the series<ref>[https://dlmf.nist.gov/10.8#E1 NIST Digital Library of Mathematical Functions], (10.8.1). Accessed on line Oct. 25, 2016.</ref>
<math display="block">Y_n(z) =-\frac{\left(\frac{z}{2}\right)^{-n}}{\pi}\sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!}\left(\frac{z^2}{4}\right)^k +\frac{2}{\pi} J_n(z) \ln \frac{z}{2} -\frac{\left(\frac{z}{2}\right)^n}{\pi}\sum_{k=0}^\infty (\psi(k+1)+\psi(n+k+1)) \frac{\left(-\frac{z^2}{4}\right)^k}{k!(n+k)!}</math>
where <math>\psi(z)</math> is the [[digamma function]], the [[logarithmic derivative]] of the [[gamma function]].<ref name="MathWorld">{{MathWorld|id=BesselFunctionoftheSecondKind|title=Bessel Function of the Second Kind}}</ref>

There is also a corresponding integral formula (for {{math|Re(''x'') > 0}}):<ref name="p. 178">Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178 p. 178].</ref>
<math display="block">Y_n(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left(e^{nt} + (-1)^n e^{-nt} \right) e^{-x \sinh t} \, dt.</math>

In the case where {{math|''n'' {{=}} 0}}: (with <math>\gamma</math> being [[Euler's constant]])<math display="block">Y_{0}\left(x\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(x\cos\theta\right)\left(\gamma+\ln\left(2x\sin^2\theta\right)\right)\, d\theta.</math>

[[File:Besselyhalf.png|thumb|300px|Plot of the Bessel function of the second kind <math>Y_\alpha(z)</math> with <math>\alpha = 0.5</math> in the complex plane from <math> -2 -2i</math> to <math>2 + 2i</math>.]]
{{math|''Y<sub>α</sub>''(''x'')}} is necessary as the second linearly independent solution of the Bessel's equation when {{mvar|α}} is an integer. But {{math|''Y<sub>α</sub>''(''x'')}} has more meaning than that. It can be considered as a "natural" partner of {{math|''J<sub>α</sub>''(''x'')}}. See also the subsection on Hankel functions below.

When {{mvar|α}} is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
<math display="block">Y_{-n}(x) = (-1)^n Y_n(x).</math>

Both {{math|''J<sub>α</sub>''(''x'')}} and {{math|''Y<sub>α</sub>''(''x'')}} are [[holomorphic function]]s of {{mvar|x}} on the [[complex plane]] cut along the negative real axis. When {{mvar|α}} is an integer, the Bessel functions {{mvar|J}} are [[entire function]]s of {{mvar|x}}. If {{mvar|x}} is held fixed at a non-zero value, then the Bessel functions are entire functions of {{mvar|α}}.

The Bessel functions of the second kind when {{mvar|α}} is an integer is an example of the second kind of solution in [[Fuchs's theorem]].