Abelian integral

The abelian integrals is an integral with the integrand, which has a certain shape. Named this integral expressions are named after the mathematician Niels Henrik Abel; they are examined especially in function theory or in algebraic geometry.

Definition

Be a rational function in two variables. Then the Abelian integral is an integral expression of the form

Where an algebraic function of is. The value of the integral depends generally on the choice of the curve, which connects with.

In algebraic or complex geometry is generalized these integral expressions with the help of rational differential forms on compact Riemannian surfaces. One speaks of a Abelian integral of the first kind if the differential form is holomorphic, of the second kind, if all poles are greater than or equal to two of the order, and else of the Third Kind

These integrals are a generalization of the known from the theory of functions, elliptic integrals. These are obtained for the special case with a polynomial of third or fourth degree without multiple roots.