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Jacobi Function of the Second Kind


\begin{displaymath}
Q_n^{(\alpha,\beta)}(x)=2^{-n-1}(x-1)^{-\alpha}(x+1)^{-\beta}\int_{-1}^1 (1-t)^{n+\alpha}(1+t)^{n+\beta}(x-t)^{-n-1}\,dt.
\end{displaymath} (1)

In the exceptional case $n=0$, $\alpha+\beta+1=0$, a nonconstant solution is given by


\begin{displaymath}
Q^{(\alpha)}(x)=\ln(x+1)+\pi^{-1}\sin(\pi\alpha)(x-1)^{-\alp...
...a}\int_{-1}^1 {(1-t)^\alpha(1+t)^\beta\over x-t} \ln(1+t)\,dt.
\end{displaymath} (2)


References

Szegö, G. ``Jacobi Polynomials.'' Ch. 4 in Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 73-79, 1975.




© 1996-9 Eric W. Weisstein
1999-05-25