查看“︁连带勒让德函数”︁的源代码

'''连带勒让德函数'''是[[连带勒让德多项式]]的推广。

下列连带勒让德方程的解，称为连带勒让德函数
:<math>(1-x^2)\,y'' -2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right]\,y = 0,\,</math>

==第一类连带勒让德 函数==
:<math>P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}),\qquad \text{for } \  |1-z|<2</math>

[[File:Associated LegenderP2d half.gif|thumb|Associated  Legendre P function]]
[[File:Associated LegenderP3d.gif|thumb|Associated  Legendre P function]]

==第二类连带勒让德函数==
:<math>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:Associated LegendreQ3D1.png|thumb|Associated  Legendre Q function]]
[[File:Associated LegenderQ2d4.gif|thumb|Associated  Legendre Q function]]
[[File:Associated LegenderQ2d3.gif|thumb|Associated  Legendre Q function]]
[[File:Associated LegenderQ2d2.gif|thumb|Associated  Legendre Q function]]
[[File:Associated LegenderQ2d1.gif|thumb|Associated  Legendre Q function]]

==参考文献==
<references/>
* {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publisher, Inc|publication-place=New York}}.
*{{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
*{{eom|id=L/l058030|first=A.B.|last= Ivanov}}
*{{Citation | last1=Snow | first1=Chester | title=Hypergeometric and Legendre functions with applications to integral equations of potential theory | url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 | publisher=U. S. Government Printing Office | location=Washington, D.C. | series=National Bureau of Standards Applied Mathematics Series, No. 19 | mr=0048145 | year=1952|origyear=1942}}
*{{Citation | last1=Whittaker | first1=E. T. |authorlink1=E. T. Whittaker| last2=Watson | first2=G. N. | authorlink2=G. N. Watson|title=A Course in Modern Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-58807-2 | year=1963 }}

[[Category:超幾何函數]]