Abelian integral

The Abelian integral is an integral with an integrand that has a certain shape. These integral expressions are named after the mathematician Niels Henrik Abel ; they are particularly studied in function theory or in algebraic geometry .

definition

Let be a rational function in two variables. Then the abelian integral is an integral term of the form ${\displaystyle R}$

${\displaystyle \int _{a}^{b}R(z,w(z))\,\mathrm {d} z,}$

where is an algebraic function of . The value of the integral generally depends on the choice of the curve from which to connect. ${\displaystyle w}$ ${\displaystyle z}$ ${\displaystyle a}$ ${\displaystyle b}$

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

These integrals are a generalization of the elliptic integrals known from function theory . This is obtained for the special case with a third or fourth degree polynomial without multiple zeros . ${\displaystyle \textstyle w(z)={\sqrt {P(z)}}}$ ${\displaystyle P(z)}$

literature

• P. Griffiths, J. Harris: Principles of Algebraic Geometry . Springer, Berlin 1994, ISBN 0-471-05059-8 (English).
• C. Neumann : Lectures on Riemann's theory of Abel integrals . 2nd Edition. BG Teubner, Leipzig 1884.

Web links

• Abelian integrals and Abelian functions
• Springer Online Reference Abelian Integral