# GenusZeroIntegration(R, F, L)¶

This internal package rationalises integrands on curves of the form: y\^2 = a x\^2 + b x + c y\^2 = (a x + b) / (c x + d) f(x, y) = 0 where f has degree 1 in x The rationalization is done for integration, limited integration, extended integration and the risch differential equation.

lift: (SparseUnivariatePolynomial F, Kernel F) -> SparseUnivariatePolynomial Fraction SparseUnivariatePolynomial F
lift(u, k) undocumented
multivariate: (SparseUnivariatePolynomial Fraction SparseUnivariatePolynomial F, Kernel F, F) -> F
multivariate(u, k, f) undocumented
palgint0: (F, Kernel F, Kernel F, F, SparseUnivariatePolynomial F) -> IntegrationResult F
palgint0(f, x, y, d, p) returns the integral of f(x, y)dx where y is an algebraic function of x satisfying d(x)\^2 y(x)\^2 = P(x).
palgint0: (F, Kernel F, Kernel F, Kernel F, F, Fraction SparseUnivariatePolynomial F, F) -> IntegrationResult F
palgint0(f, x, y, z, t, c) returns the integral of f(x, y)dx where y is an algebraic function of x satisfying x = eval(t, z, ry) and c = d/dz t; r is rational function of x, c and t are rational functions of z. Argument z is a dummy variable not appearing in f(x, y).
palgLODE0: (L, F, Kernel F, Kernel F, F, SparseUnivariatePolynomial F) -> Record(particular: Union(F, failed), basis: List F) if L has LinearOrdinaryDifferentialOperatorCategory F
palgLODE0(op, g, x, y, d, p) returns the solution of op f = g. Argument y is an algebraic function of x satisfying d(x)\^2y(x)\^2 = P(x).
palgLODE0: (L, F, Kernel F, Kernel F, Kernel F, F, Fraction SparseUnivariatePolynomial F, F) -> Record(particular: Union(F, failed), basis: List F) if L has LinearOrdinaryDifferentialOperatorCategory F
palgLODE0(op, g, x, y, z, t, c) returns the solution of op f = g. Argument y is an algebraic function of x satisfying x = eval(t, z, ry) and c = d/dz t; r is rational function of x, c and t are rational functions of z.
palgRDE0: (F, F, Kernel F, Kernel F, (F, F, Symbol) -> Union(F, failed), F, SparseUnivariatePolynomial F) -> Union(F, failed)
palgRDE0(f, g, x, y, foo, d, p) returns a function z(x, y) such that dz/dx + n * df/dx z(x, y) = g(x, y) if such a z exists, and “failed” otherwise. Argument y is an algebraic function of x satisfying d(x)\^2y(x)\^2 = P(x). Argument foo, called by foo(a, b, x), is a function that solves du/dx + n * da/dx u(x) = u(x) for an unknown u(x) not involving y.
palgRDE0: (F, F, Kernel F, Kernel F, (F, F, Symbol) -> Union(F, failed), Kernel F, F, Fraction SparseUnivariatePolynomial F, F) -> Union(F, failed)
palgRDE0(f, g, x, y, foo, t, c) returns a function z(x, y) such that dz/dx + n * df/dx z(x, y) = g(x, y) if such a z exists, and “failed” otherwise. Argument y is an algebraic function of x satisfying x = eval(t, z, ry) and c = d/dz t; r is rational function of x, c and t are rational functions of z. Argument foo, called by foo(a, b, x), is a function that solves du/dx + n * da/dx u(x) = u(x) for an unknown u(x) not involving y.
rationalize_ir: (IntegrationResult F, Kernel F) -> IntegrationResult F
rationalize_ir(irf, k1) eliminates square root k1 from the integration result.
univariate: (F, Kernel F, Kernel F, SparseUnivariatePolynomial F) -> SparseUnivariatePolynomial Fraction SparseUnivariatePolynomial F
univariate(f, k, k, p) undocumented