Path: news.mathworks.com!newsfeed-00.mathworks.com!news.kjsl.com!newsfeed.stanford.edu!newshub.sdsu.edu!flph200.ffdc.sbc.com!prodigy.net!flph199.ffdc.sbc.com!prodigy.com!flpi107.ffdc.sbc.com!flpi144.ffdc.sbc.com.POSTED!ffbda4aa!not-for-mail Reply-To: "Nasser Abbasi" <nma@12000.org> From: "Nasser Abbasi" <nma@12000.org> Newsgroups: comp.soft-sys.matlab References: <gl8cd6$4rj$1@fred.mathworks.com> Subject: Re: elliptic integral strangeness Lines: 67 X-Priority: 3 X-MSMail-Priority: Normal X-Newsreader: Microsoft Outlook Express 6.00.2900.3138 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3350 X-RFC2646: Format=Flowed; Original Message-ID: <y9Sdl.5462$jZ1.5236@flpi144.ffdc.sbc.com> NNTP-Posting-Host: 75.38.110.8 X-Complaints-To: abuse@prodigy.net X-Trace: flpi144.ffdc.sbc.com 1232596702 ST000 75.38.110.8 (Wed, 21 Jan 2009 22:58:22 EST) NNTP-Posting-Date: Wed, 21 Jan 2009 22:58:22 EST Organization: at&t http://my.att.net/ X-UserInfo1: OXZUSWKOG@C_GFLYZBNXKQ@@AZJTB_LILIXNMVMHQYUJUZ]CCVWCPG[YMDXZH^[K[FFQZHBM@FX\NJOCW^TGNQLFRFU_HSDIHX[FCUWCXLP@PBL\BKFXXVGCM\CCKFVL_T[GJLBM@Q^]WKGS]T]M^NG_YKYVGV_IJYXS@MCBT[@JPRXECDFZMSXG]NVQQTJL Date: Wed, 21 Jan 2009 20:00:19 -0800 Xref: news.mathworks.com comp.soft-sys.matlab:513072 "Matthew " <mwk5v@virginia.edu> wrote in message news:gl8cd6$4rj$1@fred.mathworks.com... > Hello everyone, > > I'm trying to evaluate a symbolic integral in Matlab whose solution is a > linear combination of two elliptic integrals, but Matlab does not appear > to evaluate the integral correctly. The integral that I am trying to > evaluate is: > > E = sin^2(x)*sqrt(sin^2(x) + m*cos^2(x)), 0 <= m <= 1 > > I want to integrate with respect to x, for x = 0..pi. > > At the limits of m, the solution is in closed-form and becomes a trivial > integral > > m=0: E = sin^2(x)*sqrt(sin^2(x)) = sin^3(x), whose integral from > x=0..pi is simply 4/3. > > m=1: E = sin^2(x)*sqrt[sin^2(x)+cos^2(x)] = sin^2(x) since > sin^2(x)+cos^2(x)=1, and the integral is simply pi/2. > > However, when I take the integral in Matlab, I get the following > expression: > > int(E) = 2/3*[m*K(g)+(m-2)*E(g)]/(m-1), g=sqrt(1-m) > > where K(g) is the complete integral of the first kind, and E(g) is the > complete integral of the second kind. Substituting m=0 into this > expression gives the correct value for int(E) = 4/3, but substituting > m=0.99999999999 into this expression (since m-1 would yield a division by > zero error), gives a very large value. Indeed, as m->1, int(E) -> -Inf. I > haven't been able to get anywhere on this problem for the last few days, > and any assistance the community can provide would be greatly appreciated. > > Thanks, > Matt > Hi Mathews; I could confirm your results using Mathematica 7 as well: Mathematica 7.0 for Students: Microsoft Windows (32-bit) Version In[1]:= r = Assuming[Element[m, Reals] && 0 <= m <= 1, Integrate[Sin[x]^2*Sqrt[Sin[x]^2 + m*Cos[x]^2], {x, 0, Pi}]]; In[2]:= Limit[r, m -> 0] 4 Out[2]= - 3 In[3]:= Limit[r, m -> 1] Pi Out[3]= -- 2 If you are not getting this using Matlab symbolic toolbox, may be you should report it to Mathworks with version number and commands used. --Nasser