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).
