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An Abelian integral, are also called a hyperelliptic integral, is an integral of the form int_0^x(dt)/(sqrt(R(t))), where R(t) is a polynomial of degree >4.

The orthogonal polynomials on the interval [-1,1] associated with the weighting functions w(x) = (1-x^2)^(-1/2) (1) w(x) = (1-x^2)^(1/2) (2) w(x) = sqrt((1-x)/(1+x)), (3) ...

Every continuous map f:S^n->R^n must identify a pair of antipodal points.

Given a Jacobi amplitude phi in an elliptic integral, the argument u is defined by the relation phi=am(u,k). It is related to the elliptic integral of the first kind F(u,k) ...

A parameter n used to specify an elliptic integral of the third kind Pi(n;phi,k).

An alternative name for an associated Legendre polynomial.

The Jack polynomials are a family of multivariate orthogonal polynomials dependent on a positive parameter alpha. Orthogonality of the Jack polynomials is proved in Macdonald ...

If xsinalpha=sin(2beta-alpha), then (1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=2int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).

Let F be a differential field with constant field K. For f in F, suppose that the equation g^'=f (i.e., g=intf) has a solution g in G, where G is an elementary extension of F ...

Given a elliptic modulus k in an elliptic integral, the modular angle alpha is defined by k=sinalpha. An elliptic integral is written I(phi|m) when the parameter m is used, ...