info prev up next book cdrom email home

Jacobi Differential Equation


\begin{displaymath}
(1-x^2)y''+[\beta-\alpha-(\alpha+\beta+2)x]y'+n(n+\alpha+\beta+1)y=0
\end{displaymath} (1)

or


\begin{displaymath}
{d\over dx}[(1-x)^{\alpha+1}(1+x)^{\beta+1}y']+n(n+\alpha+\beta+1)(1-x)^\alpha(1+x)^\beta y=0.
\end{displaymath} (2)

The solutions are Jacobi Polynomials. They can be transformed to


\begin{displaymath}
{d^2u\over dx^2}+\left[{{1\over 4}{1-\alpha^2\over(1-x)^2}+{...
...extstyle{1\over 2}}(\alpha+1)(\beta+1)\over 1-x^2}}\right]u=0,
\end{displaymath} (3)

where
\begin{displaymath}
u=u(x)=(1-x)^{(\alpha+1)/2}(1+x)^{(\beta+1)/2}P_n^{(\alpha,\beta)}(x),
\end{displaymath} (4)

and


\begin{displaymath}
{d^2u\over d\theta^2}+\left[{{{\textstyle{1\over 4}}-\alpha^...
...theta)}+\left({n+{\alpha+\beta+1\over 2}}\right)^2}\right]u=0,
\end{displaymath} (5)

where
\begin{displaymath}
u=u(\theta)=\sin^{\alpha+1/2}({\textstyle{1\over 2}}\theta)\...
...\textstyle{1\over 2}}\theta) P_n^{(\alpha,\beta)}(\cos\theta).
\end{displaymath} (6)




© 1996-9 Eric W. Weisstein
1999-05-25