連帶勒讓德函數

連帶勒讓德函數是連帶勒讓德多項式的推廣。

下列連帶勒讓德方程的解，稱為連帶勒讓德函數

$(1-x^{2})\,y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]\,y=0,\,$

第一類連帶勒讓德 函數

$P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}),\qquad {\text{for }}\ |1-z|<2$

Associated Legendre P function

Associated Legendre P function

第二類連帶勒讓德函數

$Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.$

Associated Legendre Q function

Associated Legendre Q function

Associated Legendre Q function

Associated Legendre Q function

Associated Legendre Q function

參考文獻

• Courant, Richard; Hilbert, David, Methods of Mathematical Physics, Volume 1, New York: Interscience Publisher, Inc, 1953 .
• Dunster, T. M., Legendre and Related Functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248 
• Ivanov, A.B., L/l058030, Hazewinkel, Michiel (編), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4 
• Snow, Chester, Hypergeometric and Legendre functions with applications to integral equations of potential theory, National Bureau of Standards Applied Mathematics Series, No. 19, Washington, D.C.: U. S. Government Printing Office, 1952 [1942], MR 0048145 
• Whittaker, E. T.; Watson, G. N., A Course in Modern Analysis, Cambridge University Press, 1963, ISBN 978-0-521-58807-2