Abel's addition theorem reduces a sum of integrals of a fixed algebraic function to a bounded number
of integrals of the same function, together with an
elementary function of the original endpoints.
If the associated algebraic curve has curve
genus 
,
then, modulo periods arising from closed integration paths,
 |
where 
is an Abelian integral, the 
are algebraic functions
of the 
,
and 
is a combination of algebraic functions and
their logarithms (Kleiman 2004). Choices of paths and
branches are understood in this statement.
For elliptic integrals, 
, so only one integral remains,
as in Euler's addition theorem. The related
Abel's curve theorem concerns sums taken over
intersections with a varying auxiliary algebraic curve
(Kleiman 2004).
