In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. Suppose given a Riemann surface S and on it a differential 1-form ω that is everywhere on S holomorphic, and fixing a point P on S from which to integrate. We can regard

as a multi-valued function f(Q), or (better) an honest function of the chosen path C drawn on S from P to Q. Since S will in general be multiply-connected, one should specify C, but the value will in fact only depend on the homology class, of C modulo cycles on S.

In the case of S a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as f.

Such functions were first introduced to study hyperelliptic integrals, i.e. for the case where S is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions √A, where A is a polynomial of degree > 4. The first major insights of the theory were given by Niels Abel; it was later formulated in terms of the Jacobian variety J(S). Choice of P gives rise to a standard holomorphic mapping

S → J(S)

of complex manifolds. It has the defining property that the holomorphic 1-forms on J(S), of which there are g independent ones if g is the genus of S, pull back to a basis for the differentials of the first kind on S.