In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...mathematics, an abelian integral in In mathematics, particularly in complex analysis, a Riemann surface is a one_dimensional complex manifold. ...Riemann surface theory is a function related to the In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...indefinite integral of a In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic geometry), for everywhere_regular differential 1_forms. ...differential of the first kind. Suppose given a Riemann surface S and on it a A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...differential 1_form ω that is everywhere on S Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex_differentiable at every point. ...holomorphic, and fixing a point P on S from which to integrate. We can regard

as a This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...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 A geometrical object is called simply connected if it consists of one piece and doesnt have any circle_shaped holes or handles. Higher_dimensional holes are allowed. ...multiply_connected, one should specify C, but the value will in fact only depend on the In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...homology class, of C The word modulo is the Latin ablative of modulus. ...modulo cycles on S.

In the case of S a compact Riemann surface of In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...genus 1, i.e. an In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. ...elliptic curve, such functions are the In integral calculus, an elliptic integral is any function f which can be expressed in the form where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c...elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as f.

Such functions were first introduced to study In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic geometry), for everywhere-regular differential 1-forms. ...hyperelliptic integrals, i.e. for the case where S is a In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non_trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic...algebraic functions √A, where A is a In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...polynomial of degree > 4. The first major insights of the theory were given by Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...Niels Abel; it was later formulated in terms of the For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...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, This article discusses the pullback in differential geometry. ...pull back to a basis for the differentials of the first kind on S.