Editing Legendre function (section)

== Solutions of the differential equation ==
Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the [[hypergeometric function]], <math> _2F_1</math>. With <math>\Gamma</math> being the [[gamma function]], the first solution is
<math display="block">P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{z+1}{z-1}\right]^{\mu/2} \,_2F_1 \left(-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}\right),\qquad \text{for } \  |1-z|<2,</math>
and the second is
<math display="block">Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ |z|>1.</math>
[[File:Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Legendre function of the second kind Q n(x) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if {{math|''μ''}} is non-zero. A useful relation between the {{math|''P''}} and {{math|''Q''}} solutions is [[Whipple formulae|Whipple's formula]].

===Positive integer order ===
For positive integer <math> \mu = m \in \N^+ </math> the evaluation of <math> P^\mu_\lambda </math> above involves cancellation of singular terms. We can find the limit valid for <math> m \in \N_0 </math> as<ref>{{cite journal | url=https://www.degruyter.com/document/doi/10.1515/mcma-2018-0001/html?lang=de | doi=10.1515/mcma-2018-0001 | title=Fast generation of isotropic Gaussian random fields on the sphere | date=2018 | last1=Creasey | first1=Peter E. | last2=Lang | first2=Annika | journal=Monte Carlo Methods and Applications | volume=24 | issue=1 | pages=1–11 | arxiv=1709.10314 | bibcode=2018MCMA...24....1C | s2cid=4657044 }}</ref>

<math display="block">P^m_\lambda(z) = \lim_{\mu \to m} P^\mu_\lambda (z) = \frac{(-\lambda )_m (\lambda + 1)_m}{m!} \left[\frac{1-z}{1+z}\right]^{m/2} \,_2F_1 \left(-\lambda, \lambda+1; 1+m; \frac{1-z}{2}\right), </math>

with <math>(\lambda)_{n}</math> the (rising) [[Pochhammer symbol]].