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In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
where R {\displaystyle R} is an arbitrary rational function of the two variables x {\displaystyle x} and w {\displaystyle w} , which are related by the equation
where F {\displaystyle F} is an irreducible polynomial in w {\displaystyle w} ,
whose coefficients φ j {\displaystyle \varphi _{j}} , j = 0 , 1 , … , n {\displaystyle j=0,1,\ldots ,n} are rational functions of x {\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 z {\displaystyle z}.