Uniform Expansions of the Third Symmetric Standard Elliptic Integral

Abstract We consider the third symmetric standard elliptic integral $$R_J(x,y,z,p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$R_J(x,y,z,p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$R_F(x,y,z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$R_D(x,y,z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>D</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . These expansions hold uniformly for one or two of the variables in large closed unbounded subsets of $$\mathbb {C}\setminus (-\infty ,0]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>\</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . These expansions are accompanied by error bounds, and their accuracy and uniform features are illustrated by means of some numerical experiments.