Jacobi Function of the Second Kind

A Jacobi function of the second kind is defined by

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.

In the exceptional case , , a nonconstant solution is given by

Q^((alpha))(x)=ln(x+1)+pi^(-1)sin(pialpha)(x-1)^(-alpha)(x+1)^(-beta)
×int_(-1)^1((1-t)^alpha(1+t)^beta)/(x-t)ln(1+t)dt.

See also

Jacobi Differential Equation, Jacobi Polynomial

Related Wolfram sites

https://functions.wolfram.com/HypergeometricFunctions/JacobiPGeneral/

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References

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

Referenced on Wolfram|Alpha

Jacobi Function of the Second Kind

Cite this as:

Weisstein, Eric W. "Jacobi Function of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobiFunctionoftheSecondKind.html

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