From: rusin@shuksan.math.niu.edu (Dave Rusin) Subject: Another Maple integration gaffe Date: 10 Mar 1999 09:29:12 GMT Newsgroups: sci.math.symbolic The integration command int(ln(U)^(1/2)/U/(-1+U)^(1/2),U); gives an erroneous result in release 4; I don't have release 5 to test. The function and "its antiderivative" are 1/2 3/2 ln(U) ln(U) -------------, 2 ----------- 1/2 1/2 U (-1 + U) (-1 + U) Setting infolevel[int]:=5: reveals that Maple attempts to perform integration by parts, and then decides that the resulting second integrand is zero. Differentiating the wrong answer and asking Maple to integrate that derivative leaves it stumped. I had always wanted my calculus students to have the same skills as Maple; unfortunately I seem to have gotten my wish :-) Interestingly, the wrong answer happens to be the right answer to the function I really wanted to integrate in the first place! dave ============================================================================== From: Helmut Kahovec Subject: Re: Another Maple integration gaffe Date: Thu, 11 Mar 1999 05:55:50 +0100 Newsgroups: [missing] To: Dave Rusin Dave Rusin wrote: > > The integration command > int(ln(U)^(1/2)/U/(-1+U)^(1/2),U); > gives an erroneous result in release 4; I don't have release 5 to test. > > The function and "its antiderivative" are > 1/2 3/2 > ln(U) ln(U) > -------------, 2 ----------- > 1/2 1/2 > U (-1 + U) (-1 + U) > > Setting infolevel[int]:=5: reveals that Maple attempts to perform > integration by parts, and then decides that the resulting second > integrand is zero. Differentiating the wrong answer and asking Maple > to integrate that derivative leaves it stumped. > Dave, In my last email to you I wrote: > I also do not understand the theory behind those algorithms. In fact I meant: > As to those algorithms, I am not an expert either. I apologize for having appeared somewhat impolite. Additionally, I referred to Release 5 in that email. As to your current problem, Release 4 cannot compute the antiderivative of ln(U)^(1/2)/U/(-1+U)^(1/2) but Release 5 can. However, we must help Maple a bit: > restart; > j:=ln(U)^(1/2)/U/(-1+U)^(1/2); sqrt(ln(U)) j := -------------- U sqrt(-1 + U) > rU:=U=exp(V^2); 2 rU := U = exp(V ) > rV:=ln(U)^(1/2)=V; rV := sqrt(ln(U)) = V > j1:=subs({rU,rV},j)*diff(subs(rU,U),V); 2 V j1 := 2 ------------------ 2 sqrt(-1 + exp(V )) > J1:=int(j1,V); /infinity | ----- | \ J1 := 2 sqrt(2) | ) GAMMA(_k1 + 1/2) ( | / | ----- \_k1 = 0 2 - 1/2 V sqrt(2) sqrt(2 _k1 + 1) exp(- 1/2 V (2 _k1 + 1)) / + 1/2 sqrt(Pi) erf(1/2 sqrt(2) V sqrt(2 _k1 + 1))) / ( / \ | (3/2) | (2 _k1 + 1) GAMMA(_k1 + 1))|/sqrt(Pi) | | / > simplify(j1-value(diff(J1,V)),assume=real); 0 Helmut ============================================================================== From: "Marc A. Murison" Subject: Re: Another Maple integration gaffe Date: Fri, 12 Mar 1999 11:02:47 -0500 Newsgroups: sci.math.symbolic To: Dave Rusin > restart; > interface(version); Maple Worksheet Interface, Release 5.1, IBM INTEL NT, Nov 5 1998 > infolevel[int] := 1: > int(ln(U)^(1/2)/(U*(-1+U)^(1/2)),U); int/indef1: first-stage indefinite integration int/indef2: second-stage indefinite integration int/indef2: applying algebraic substitution int/indef1: first-stage indefinite integration int/indef1: first-stage indefinite integration int/indef2: second-stage indefinite integration int/ln: case of integrand containing ln int/rischnorm: enter Risch-Norman integrator int/rischnorm: exit Risch-Norman integrator int/risch: enter Risch integration int/risch: exit Risch integration (3/2) ln(U) 2 ------------ sqrt(-1 + U) Dave Rusin wrote: [original post deleted -- djr] -- Marc A. Murison Astronomical Applications Dept. U.S. Naval Observatory, Washington, D.C. http://aa.usno.navy.mil/murison/ mailto:murison@aa.usno.navy.mil Utinam logica falsa tuam philosophiam totam suffodiant!