Abelian integral

In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form

${\displaystyle \int _{z_{0}}^{z}R(x,w)\,dx,}$

where ${\displaystyle R(x,w)}$ is an arbitrary rational function of the two variables ${\displaystyle x}$ and ${\displaystyle w}$, which are related by the equation

${\displaystyle F(x,w)=0,}$

where ${\displaystyle F(x,w)}$ is an irreducible polynomial in ${\displaystyle w}$,

${\displaystyle F(x,w)\equiv \varphi _{n}(x)w^{n}+\cdots +\varphi _{1}(x)w+\varphi _{0}\left(x\right),}$

whose coefficients ${\displaystyle \varphi _{j}(x)}$, ${\displaystyle j=0,1,\ldots ,n}$ are rational functions of ${\displaystyle x}$. The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of ${\displaystyle z}$.

Abelian integrals are natural generalizations of elliptic integrals, which arise when

${\displaystyle F(x,w)=w^{2}-P(x),\,}$

where ${\displaystyle P\left(x\right)}$ is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where ${\displaystyle P(x)}$, in the formula above, is a polynomial of degree greater than 4.