Euler's addition theorem for elliptic integrals states that, with 
,
 |
where
 |
For real variables with 
and 
, use positive square
roots and require the sum of the two integrals to
lie between 
and 
,
where 
is the complete elliptic integral
of the first kind. The formula follows from the addition identity for the Jacobi elliptic function 
(DLMF 22.8.1).
