Year: 2026
Type: article
Abstract: We consider the third symmetric standard elliptic integral $$R_J(x,y,z,p)$$ R J ( x , y , z , p ) for complex values of its variables. By homogeneity arguments, this function is indeed a function of only three variables, and we derive two different integral representations of $$R_J(x,y,z,p)$$ R J ( x , y , z , p ) which only involve three variables. Both integral representations are suitable for the analysis introduced in [Lopez, Pagola and Palacios, 2021] to derive uniform expansions of parametric integrals. Using this theory, we derive six convergent expansions of this function in terms of elementary functions; two of these expansions also involve the other two symmetric standard elliptic integrals $$R_F(x,y,z)$$ R F ( x , y , z ) and $$R_D(x,y,z)$$ R D ( x , y , z ) . These expansions hold uniformly for one or two of the variables in large closed unbounded subsets of $$\mathbb {C}\setminus (-\infty ,0]$$ C \ ( - ∞ , 0 ] . These expansions are accompanied by error bounds, and their accuracy and uniform features are illustrated by means of some numerical experiments.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7147595857