Re: Help to solve an integral by using Mathematica Integrate[Sqrt[t
- To: mathgroup at smc.vnet.net
- Subject: [mg112819] Re: Help to solve an integral by using Mathematica Integrate[Sqrt[t
- From: Valeri Astanoff <astanoff at gmail.com>
- Date: Fri, 1 Oct 2010 06:02:40 -0400 (EDT)
- References: <i84adn$gvn$1@smc.vnet.net>
On 1 oct, 11:42, Leonid Shifrin <lsh... at gmail.com> wrote: > To my surprise, Mathematica has been very reluctant to help here, despite > the > seemingly simple form of the integral. Here is what I did anyway: > > 1. Find an indefinite integral: > > In[94]:= expr = Integrate[Sqrt[t (1 - t) (z - t)], t] > > Out[94]= (Sqrt[(-1 + t) t (t - z)] (2 t (-1 + 3 t - z) + ( > 2 (-1 + t) (-((2 t (t - z) (1 - z + z^2))/(-1 + t)^2) - ( > 2 I Sqrt[t/(-1 + t)] Sqrt[( > t - z)/(-1 + t)] (1 - z + z^2) EllipticE[ > I ArcSinh[1/Sqrt[-1 + t]], 1 - z])/Sqrt[-1 + t] + ( > I Sqrt[t/(-1 + t)] Sqrt[(t - z)/(-1 + t)] > z (1 + z) EllipticF[I ArcSinh[1/Sqrt[-1 + t]], 1 - z])/ > Sqrt[-1 + t]))/(t - z)))/(15 t) > > 2. Find a limit on the lower and (at zero): > > In[95]:= expr0 = Limit[expr , t -> 0] > > Out[95]= -(2/ 15) I (2 (1 - z + z^2) EllipticE[1 - z] - > z (1 + z) EllipticK[1 - z]) > > 3. Make a substitution z->t-q in the indefinite integral result: > > In[96]:= expr1 = expr /. z -> t - q > > Out[96]= (Sqrt[ > q (-1 + t) t] (2 t (-1 + q + 2 t) + ( > 2 (-1 + t) (-((2 q t (1 + q - t + (-q + t)^2))/(-1 + t)^2) - ( > 2 I Sqrt[q/(-1 + t)] Sqrt[ > t/(-1 + t)] (1 + q - t + (-q + t)^2) EllipticE[ > I ArcSinh[1/Sqrt[-1 + t]], 1 + q - t])/Sqrt[-1 + t] + ( > I Sqrt[q/(-1 + t)] Sqrt[ > t/(-1 + t)] (-q + t) (1 - q + t) EllipticF[ > I ArcSinh[1/Sqrt[-1 + t]], 1 + q - t])/Sqrt[-1 + t]))/ > q))/(15 t) > > 4. Simplify it: > In[97]:= expr2 = FullSimplify[expr1, t > 0 && t < 1 && q > 0 && q < 1= ] > > Out[97]= (Sqrt[ > q (-1 + t) t] (-((2 t (1 + q (3 + 2 q - 5 t) + t))/(-1 + t)) - ( > 2 I t (2 (1 + q + q^2 - 2 q t + (-1 + t) t) EllipticE[ > I ArcCoth[Sqrt[t]], > 1 + q - t] - (-1 + q - t) (q - t) EllipticF[ > I ArcCoth[Sqrt[t]], 1 + q - t]))/Sqrt[q (-1 + t) t]))/(15= t) > > 5. Get the result for the upper end by substituting q->0 and then t->z: > > In[98]:= expr3 = (expr2 // Apart) /. q -> 0 /. t -> z > > Out[98]= -(4/15) I EllipticE[I ArcCoth[Sqrt[z]], 1 - z] - > 2/15 I z^2 (2 EllipticE[I ArcCoth[Sqrt[z]], 1 - z] - > EllipticF[I ArcCoth[Sqrt[z]], 1 - z]) + > 2/15 I z (2 EllipticE[I ArcCoth[Sqrt[z]], 1 - z] + > EllipticF[I ArcCoth[Sqrt[z]], 1 - z]) > > 6. Get the final result: > > In[99]:= expr4 = FullSimplify[expr3 - expr0] > > Out[99]= > 2/15 I (2 (1 + (-1 + z) z) EllipticE[1 - z] - > 2 (1 + (-1 + z) z) EllipticE[I ArcCoth[Sqrt[z]], 1 - z] + > z (1 + z) (EllipticF[I ArcCoth[Sqrt[z]], 1 - z] - > EllipticK[1 - z])) > > Now, it turns out that the sign is wrong. All my attempts to verify the > correctness of > this analytically failed (I did not try too hard though). Neither was I a= ble > to reduce it > to the manifestly real form. > > The final form of the result is then (correcting the sign and taking the > real part): > > exprint[z_] := > Re[-(2/15) > I (2 (1 + (-1 + z) z) EllipticE[1 - z] - > 2 (1 + (-1 + z) z) EllipticE[I ArcCoth[Sqrt[z]], 1 - z] + > z (1 + z) (EllipticF[I ArcCoth[Sqrt[z]], 1 - z] - > EllipticK[1 - z]))]; > > I did compare it to the result of numerical integration: > > exprintN[z_?NumericQ] := > NIntegrate[Sqrt[t (1 - t) (z - t)], {t, 0, z}] > > Plot[{exprint[z], exprintN[z]}, {z, 0, 1}] > > Plot[{exprint[z] - exprintN[z]}, {z, 0, 1}] > > And they seem to agree, but that's about all I could squeeze out of it. > > Hope this helps. > > Regards, > Leonid > > > > On Thu, Sep 30, 2010 at 12:51 PM, Hugo <hpe650... at gmail.com> wrote: > > Could anybody help me to implement this integral in Mathematica? > > Integrate[Sqrt[t (1-t) (z-t)],{t,0,z}] where z and t are real with > > intervals 0<t<1 and 0<z<1. I'd really appreciate any help for this > > problem.- Masquer le texte des messages pr=E9c=E9dents - > > - Afficher le texte des messages pr=E9c=E9dents - I suggest this form : (1/15)*(4*(1 + (z-1)*z)*EllipticE[z] - 2*(2 + (z-3)*z)*EllipticK[z]) v.a.