Bujanda Cirauqui, BlancaLópez García, José LuisPagola Martínez, Pedro JesúsPalacios Herrero, Pablo2022-09-082022-09-082022Bujanda, B., López, J.L., Pagola, P.J. et al. An Analytic Representation of the Second Symmetric Standard Elliptic Integral in Terms of Elementary Functions. Results Math 77, 171 (2022). https://doi.org/10.1007/s00025-022-01707-31422-6383 (Print)1420-9012 (Electronic)10.1007/s00025-022-01707-3https://academica-e.unavarra.es/handle/2454/43963We derive new convergent expansions of the symmetric standard elliptic integral RD(x,y,z), for x,y,z∈C∖(−∞,0], in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which RD(x,y,z) is a particular case. It turns out that this parametric integral is an integral representation of the Appell function F1(a;b,c;a+1;x,y). Therefore, as a byproduct, we deduce convergent expansions of F1(a;b,c;a+1;x,y). We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.application/pdfeng© 2022 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.Symmetric standard elliptic integralsAppell functionConvergent expansionsUniform expansionsError boundsArtículo / ArtikuluaAcceso abierto / Sarbide irekiainfo:eu-repo/semantics/openAccess