MaplePrimes - comments on Post, beta function vs. sine

MaplePrimes - comments on Post, beta function vs. sine http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 22:18:18 GMT Wed, 10 Jun 2026 22:18:18 GMT The latest comments added to the Post, beta function vs. sine http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, beta function vs. sine http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine example? http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144859 I get in Maple 15.01: <pre>int(1/(1+x^a), x=0..infinity); #no assumption on a ! Pi Pi csc(----) a ------------ a int(1/(1+x^(5/2)), x=0..infinity); Pi 2/5 --------- 2 Pi sin(----) 5 </pre> The latest comments added to the Post, beta function vs. sine 144859 Thu, 21 Mar 2013 07:49:57 Z Alejandro Jakubi Alejandro Jakubi Try 7/2 http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144862 Try 7/2. I get a Beta function in Maple 17. The latest comments added to the Post, beta function vs. sine 144862 Thu, 21 Mar 2013 08:20:50 Z Carl Love Carl Love Change http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144865 And it might be attainable in Maple, <pre>restart: e1:=int(1/(1+x^(7/2)), x=0..infinity); 2 /2 5\ - Beta|-, -| 7 \7 7/ e2:=convert(e1,Int); /1 2 | 1 - | ----------------------- d_k1 7 | (5/7) (2/7) /0 _k1 (1 - _k1) e3:=subsindets(e2, specfunc(anything,Int), z->IntegrationTools:-Change(z, IntegrationTools:-GetVariable(z)=1/x)); /infinity 2 | 1 - | -------------- dx 7 | (2/7) /1 x (x - 1) value(e3); 2 /2 \ - Pi csc|- Pi| 7 \7 / </pre>  And, FWIW... <pre>H:=Int(1/(1+x^(7/2)), x=0..infinity); /infinity | 1 | ---------- dx | (7/2) /0 1 + x evalf(%); 1.148070807 Q:=IntegrationTools:-Change(H,x=t-1): simplify(allvalues(value(Q))); 8 / /1 \ /3 \ /2 \\ - -- Pi |-2 sin|- Pi| + sin|- Pi| - 3 sin|- Pi|| 49 \ \7 / \7 / \7 // evalf(%); 1.148070807 </pre> acer The latest comments added to the Post, beta function vs. sine 144865 Thu, 21 Mar 2013 09:07:12 Z acer acer method cook http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144866 OK. It is again the issue of the first successful method in the row. For a=5/2 and a=7/2 the first successful method is cook: <pre>infolevel[IntegrationTools]:=3: int(1/(1+x^(5/2)), x=0..infinity); Definite Integration: Integrating expression on x=0..infinity Definite Integration: Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour] LookUp Integrator: unable to find the specified integral in the table Cook LookUp Integrator: returning answer from cook pattern 1a Definite Integration: Method cook succeeded. Definite Integration: Finished sucessfully. Pi 2/5 --------- 2 Pi sin(----) 5 int(1/(1+x^(7/2)), x=0..infinity); Definite Integration: Integrating expression on x=0..infinity Definite Integration: Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour] LookUp Integrator: unable to find the specified integral in the table Cook LookUp Integrator: returning answer from cook pattern 1a Definite Integration: Method cook succeeded. Definite Integration: Finished sucessfully. 2/7 Beta(2/7, 5/7) </pre> Most interesting as both results in terms of sin and Beta arise from the same pattern 1a. A workaround to get the answer in terms of sin is suppressing the method cook, so that the method meijergspecial gets its oportunity: <pre>int(1/(1+x^(7/2)), x=0..infinity,method=nocook); 3 Pi 2/7 Pi sec(----) 14 convert(%,sin); Pi 2/7 --------- 5 Pi sin(----) 7 </pre> The latest comments added to the Post, beta function vs. sine 144866 Thu, 21 Mar 2013 09:12:07 Z Alejandro Jakubi Alejandro Jakubi contour http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144869 <a href="http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine#comment144866">@Alejandro Jakubi</a> Good observation, Alejandro. The `contour` method also gets it in M16. <pre>int(1/(1+x^(7/2)), x=0..infinity, method=_RETURNVERBOSE); simplify(allvalues(%)); # to simplify the two FTOC methods </pre> The latest comments added to the Post, beta function vs. sine 144869 Thu, 21 Mar 2013 09:35:40 Z acer acer GAMMA http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine?ref=Feed:MaplePrimes:beta function vs. sine:Comments#comment144871 <a href="http://www.mapleprimes.com/posts/144855-Beta-Function-Vs-Sine#comment144869">@acer</a>  In the "end", the differences or complication with method cook can be traced to a peculiarity in the normalization of GAMMA: <pre>trace(GAMMA): Beta(2/5,3/5); Beta(2/7,5/7); </pre> This problem of the first "successful" method producing a more complicated or wrong answer does occur too frequently, as posts here and elsewhere show. A better scheme is needed beyond sequentially trying methods. The probability of a user observing the output from the method _RETURNVERBOSE is negligible. So, the potential of the definite integration methods implemented in Maple is underused. The latest comments added to the Post, beta function vs. sine 144871 Thu, 21 Mar 2013 10:10:52 Z Alejandro Jakubi Alejandro Jakubi