Abelian integral

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 form ω that is everywhere on S holomorphic, and fixing a point P on S from which to integrate. We can regard

:int_P^Q omega

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 (mathematics) 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 function (mathematics)

: 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 .