http://www.mapleprimes.com/questions/40916-Elliptic-Integrals en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Thu, 11 Jun 2026 16:24:13 GMT Thu, 11 Jun 2026 16:24:13 GMT The latest answers and comments added to the Question, Elliptic Integrals http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/questions/40916-Elliptic-Integrals http://www.mapleprimes.com/questions/40916-Elliptic-Integrals?ref=Feed:MaplePrimes:Elliptic Integrals:Comments#answer75879 If memory serves me right, the elliptic functions in their current form were introduced to Maple for the explicit purpose of making the answers from definite integration as simple as possible [somewhere between 5.2 and 5.4 I believe]. The conventions were definitely influenced by A&S, as is the convention for most of Maple's special functions, but ``tailored'' to the application of definite integration of variants of the functions that appear on ?EllipticF. Getting branch-cuts ``just right'' was a large amount of work, and the particular form was chosen to make things as simple and as uniform as possible. Some of that information should be available on the web somewhere, but I can't find it right now -- I will look again later. If memory serves me right, the elliptic functions in their current form were introduced to Maple for the explicit purpose of making the answers from definite integration as simple as possible [somewhere between 5.2 and 5.4 I believe]. The conventions were definitely influenced by A&S, as is the convention for most of Maple's special functions, but ``tailored'' to the application of definite integration of variants of the functions that appear on ?EllipticF. Getting branch-cuts ``just right'' was a large amount of work, and the particular form was chosen to make things as simple and as uniform as possible. Some of that information should be available on the web somewhere, but I can't find it right now -- I will look again later. 75879 Sun, 26 Aug 2007 18:48:33 Z JacquesC JacquesC http://www.mapleprimes.com/questions/40916-Elliptic-Integrals?ref=Feed:MaplePrimes:Elliptic Integrals:Comments#answer75878 it would be not a bad idea to add comments in the online which conventions are used for special functions ... if I remember correctly there are also differences between A & S and Gradsteyn & Ryzhik ... it would at least save the time to look it up it would be not a bad idea to add comments in the online which conventions are used for special functions ... if I remember correctly there are also differences between A & S and Gradsteyn & Ryzhik ... it would at least save the time to look it up 75878 Sun, 26 Aug 2007 20:09:52 Z Axel Vogt Axel Vogt http://www.mapleprimes.com/questions/40916-Elliptic-Integrals?ref=Feed:MaplePrimes:Elliptic Integrals:Comments#answer75854 Jacques, your memory serves you well: elliptic integrals first appeared in Maple V R4. The most noticeable differences between then and now are the help pages. Then, they were clear and referred unambiguously to A&S Chapter 17. However, they did not illustrate the trigonometric forms. To my mind, the help page in Maple 11.01 for EllipticF seems unnecessarily cluttered with the relationships between the EllipticF integral and the InverseJacobiAM, etc, functions. I would suggest they should be removed to a linked page for details, or to the existing example worksheet. Again, the help pages do not illustrate the trigonometric forms It should be fairly easy to amend the help pages and compare the elliptic integrals with A&S directly. Something on these lines might be acceptable: Maple uses k^2 as the parameter, while A&S Chapter 17 uses m so k^2 = m = sin(alpha)^2 and with the amplitude, phi, limited to 0<phi<Pi/2 the following are identities: EllipticF(sin(phi),sin(alpha)) = Int( (1 - sin(alpha)^2*sin(theta)^2)^(-1/2), theta=0..phi) Compare with A&S 17.2.6 EllipticF(x,k) = Int( (1 - t^2)^(-1/2)*(1 - k^2*t^2)^(-1/2), t=0..x) Compare with A&S 17.2.7 EllipticE(x,k) = Int( (1-t^2)^(-1/2)*(1-k^2*t^2)^(1/2),t=0..x) Compare with A&S 17.2.8 EllipticE(sin(phi),sin(alpha)) = Int(((1 - sin(theta)^2*sin(alpha)^2))^(1/2), theta=0..phi) Compare with A&S 17.2.9 EllipticPi(sin(phi),n,sin(alpha)) = Int((1 - n*sin(theta)^2)^(-1) * (1 -sin(alpha)^2*sin(theta)^2)^(-1/2), theta=0..phi) Compare with A&S 17.2.14 EllipticPi(x,n,k) = Int((1 - n*t^2)^(-1)*(1 - t^2)^(-1/2)*(1 - k^2*t^2)^(-1/2), t=0..x) Compare with A&S 17.2.15 All corrections gratefully received. J. Tarr Jacques, your memory serves you well: elliptic integrals first appeared in Maple V R4. The most noticeable differences between then and now are the help pages. Then, they were clear and referred unambiguously to A&S Chapter 17. However, they did not illustrate the trigonometric forms. To my mind, the help page in Maple 11.01 for EllipticF seems unnecessarily cluttered with the relationships between the EllipticF integral and the InverseJacobiAM, etc, functions. I would suggest they should be removed to a linked page for details, or to the existing example worksheet. Again, the help pages do not illustrate the trigonometric forms It should be fairly easy to amend the help pages and compare the elliptic integrals with A&S directly. Something on these lines might be acceptable: Maple uses k^2 as the parameter, while A&S Chapter 17 uses m so k^2 = m = sin(alpha)^2 and with the amplitude, phi, limited to 0<phi<Pi/2 the following are identities: EllipticF(sin(phi),sin(alpha)) = Int( (1 - sin(alpha)^2*sin(theta)^2)^(-1/2), theta=0..phi) Compare with A&S 17.2.6 EllipticF(x,k) = Int( (1 - t^2)^(-1/2)*(1 - k^2*t^2)^(-1/2), t=0..x) Compare with A&S 17.2.7 EllipticE(x,k) = Int( (1-t^2)^(-1/2)*(1-k^2*t^2)^(1/2),t=0..x) Compare with A&S 17.2.8 EllipticE(sin(phi),sin(alpha)) = Int(((1 - sin(theta)^2*sin(alpha)^2))^(1/2), theta=0..phi) Compare with A&S 17.2.9 EllipticPi(sin(phi),n,sin(alpha)) = Int((1 - n*sin(theta)^2)^(-1) * (1 -sin(alpha)^2*sin(theta)^2)^(-1/2), theta=0..phi) Compare with A&S 17.2.14 EllipticPi(x,n,k) = Int((1 - n*t^2)^(-1)*(1 - t^2)^(-1/2)*(1 - k^2*t^2)^(-1/2), t=0..x) Compare with A&S 17.2.15 All corrections gratefully received. J. Tarr 75854 Wed, 29 Aug 2007 16:15:29 Z Mariner Mariner http://www.mapleprimes.com/questions/40916-Elliptic-Integrals?ref=Feed:MaplePrimes:Elliptic Integrals:Comments#answer75836 Gentleman, Thank you for the support and assistance in using Maple to work with elliptic integrals. My motivation for studying this problem is to develop an understanding of the analytical approach to the structural stability of confined cylinders and compare the results to a finite element formulation. At the time of the paper there was a difference of opinion among designers as to why the buckling load produced in testing machines was significantly less than the load predicted by the closed form solution. The difference is generally thought to be the result of imperfections in the material and geometry of the cylinder. As regards the reason for using an elliptic integral in the solution of the problem, I would call your attention to Figure 4 of the Lo paper. <a href='http://www.mapleprimes.com/files/4865_Lo_p693.pdf'>Download 4865_Lo_p693.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1670'>View file details</a> As noted on page 692, the angle phi starts out equal to 0 at the origin, decreases along the curve to a maximum negative value at the inflection point, and then becomes positive from there on. To solve equation (8) the author introduces the transformation of phi shown in (9) to eliminate the change of sign. It is my understanding that this substitution is what suggests the use of the ellipic integrals to solve the equation, subject to the boundary conditions given in (11). After developing a function, f(x,y) for equation (14), I'm going to work on solving the case of ring buckling under an end compressive load (pp. 694-695) and compare the critical end compressive load as shown in Figure 9 with a FEA solution given by ANSYS. I'm sure I will have questions... <a href='http://www.mapleprimes.com/files/4865_Lo_p694.pdf'>Download 4865_Lo_p694.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1671'>View file details</a> <a href='http://www.mapleprimes.com/files/4865_Lo_p695.pdf'>Download 4865_Lo_p695.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1672'>View file details</a> I look forward to developing my analytical skills and the use of Maple. Thank you again for your assistance. Respectfully, Wayne Gentleman, Thank you for the support and assistance in using Maple to work with elliptic integrals. My motivation for studying this problem is to develop an understanding of the analytical approach to the structural stability of confined cylinders and compare the results to a finite element formulation. At the time of the paper there was a difference of opinion among designers as to why the buckling load produced in testing machines was significantly less than the load predicted by the closed form solution. The difference is generally thought to be the result of imperfections in the material and geometry of the cylinder. As regards the reason for using an elliptic integral in the solution of the problem, I would call your attention to Figure 4 of the Lo paper. <a href='http://www.mapleprimes.com/files/4865_Lo_p693.pdf'>Download 4865_Lo_p693.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1670'>View file details</a> As noted on page 692, the angle phi starts out equal to 0 at the origin, decreases along the curve to a maximum negative value at the inflection point, and then becomes positive from there on. To solve equation (8) the author introduces the transformation of phi shown in (9) to eliminate the change of sign. It is my understanding that this substitution is what suggests the use of the ellipic integrals to solve the equation, subject to the boundary conditions given in (11). After developing a function, f(x,y) for equation (14), I'm going to work on solving the case of ring buckling under an end compressive load (pp. 694-695) and compare the critical end compressive load as shown in Figure 9 with a FEA solution given by ANSYS. I'm sure I will have questions... <a href='http://www.mapleprimes.com/files/4865_Lo_p694.pdf'>Download 4865_Lo_p694.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1671'>View file details</a> <a href='http://www.mapleprimes.com/files/4865_Lo_p695.pdf'>Download 4865_Lo_p695.pdf</a><br/><a href='http://www.mapleprimes.com/viewfile/1672'>View file details</a> I look forward to developing my analytical skills and the use of Maple. Thank you again for your assistance. Respectfully, Wayne 75836 Fri, 31 Aug 2007 15:38:12 Z Wayne Wayne http://www.mapleprimes.com/questions/40916-Elliptic-Integrals?ref=Feed:MaplePrimes:Elliptic Integrals:Comments#comment85093 I believe that, to ensure that this suggestion "effectively" gets picked up, it has been recommended that you submit this directly to support@maplesoft.com (they do not have the time to gather up stuff from mapleprimes, or so I am told). I believe that, to ensure that this suggestion "effectively" gets picked up, it has been recommended that you submit this directly to support@maplesoft.com (they do not have the time to gather up stuff from mapleprimes, or so I am told). 85093 Wed, 29 Aug 2007 17:14:07 Z JacquesC JacquesC