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Re: Definite Integration in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg74667] Re: Definite Integration in Mathematica
*From*: "dimitris" <dimmechan at yahoo.com>
*Date*: Fri, 30 Mar 2007 03:01:02 -0500 (EST)
*References*: <etqo3f$10i$1@smc.vnet.net><ett73k$h3g$1@smc.vnet.net>
> Back in December, in the thread which led to this one, I gave an
> antiderivative which is continuous along the whole real line:
>
> ArcTan[(4x + Sqrt[2(15 + Sqrt[241])])/(2 - Sqrt[2(-15 + Sqrt[241])])] +
> ArcTan[(4x - Sqrt[2(15 + Sqrt[241])])/(2 + Sqrt[2(-15 + Sqrt[241])])]
Could we show me how you get this antiderivative for the integrand
f[x_] = (x^2 + 2*x + 4)/(x^4 - 7*x^2 + 2*x + 17);
BTW,
Using Mathematica I was able to get a continuous in the real axis
antiderivative as follows
Times @@ Apply[#1[[1]] - #1[[2]] & , Solve[Denominator[f[x]] == 0, x],
1];
Apart[(4 + 2*x + x^2)/%];
Map[FullSimplify, %, 1];
(Integrate[#1, x] & ) /@ %;
FF[x_] = FullSimplify[%]
FullSimplify[D[FF[x], x] == f[x]]
(1/4)*(-2*ArcTan[(2*x)/(-1 + Sqrt[-15 - 4*I])] + 2*ArcTan[(2*x)/(1 +
Sqrt[-15 - 4*I])] -
2*ArcTan[(2*x)/(-1 + Sqrt[-15 + 4*I])] + 2*ArcTan[(2*x)/(1 +
Sqrt[-15 + 4*I])] -
I*(Log[(7 + 2*I) + Sqrt[-15 - 4*I] - 2*x^2] - Log[(7 - 2*I) +
Sqrt[-15 + 4*I] - 2*x^2] +
Log[(-7 - 2*I) + Sqrt[-15 - 4*I] + 2*x^2] - Log[(-7 + 2*I) +
Sqrt[-15 + 4*I] + 2*x^2]))
True
Plot[FF[x], {x, 0, 4}]
Thanks a lot!
Dimitris
=CF/=C7 David W.Cantrell =DD=E3=F1=E1=F8=E5:
> "Michael Weyrauch" <michael.weyrauch at gmx.de> wrote:
> > Hello,
> >
> > another nice example, where the result for the integral given by
> > Mathematica just cannot be right.
>
> I understand your point, but it's impossible to have a result here which
> is "right" throughout the complex plane.
>
> > The indefinite integral of a continuuos function cannot have a jump.
>
> Again, I understand your point. You're thinking just about integrating
> along the real line; the integrand
>
> (x^2 + 2x + 4)/(x^4 - 7x^2 + 2x + 17)
>
> is indeed continuous there. But computer algebra systems sometimes give
> antiderivatives which are "right" only piecewise. You're welcome to
> consider that unfortunate in this case because we can do better; see belo=
w=2E
>
> > That -- to my opinion -- is mathematical
> > nonsens. We all learned that integration "smoothens", i.e. if the
> > integrand is somewhat "ugly" the integral is less "ugly". (Never mind my
> > English!).
> >
> > So the result should actually be presented as
> >
> > F[x_]=ArcTan[(1 + x)/(4 - x^2)]*UnitStep[2 - x] +
> > (Pi + ArcTan[(1 + x)/(4 - x^2)])*UnitStep[-2 + x]
>
> No, not if you're wanting an antiderivative valid along the whole real
> line. Your F has a jump discontinuity at x = -2. Furthermore, it is
> Indeterminate at x = +2, although that singularity is removable.
>
> Back in December, in the thread which led to this one, I gave an
> antiderivative which is continuous along the whole real line:
>
> ArcTan[(4x + Sqrt[2(15 + Sqrt[241])])/(2 - Sqrt[2(-15 + Sqrt[241])])] +
> ArcTan[(4x - Sqrt[2(15 + Sqrt[241])])/(2 + Sqrt[2(-15 + Sqrt[241])])]
>
> David W. Cantrell
>
> > This is a perfectly nice function without jumps and it is the
> > antiderivative of your integrand, and the fundamental theorem of calcul=
us
> > works with it...
> >
> > So, despite my love for Mathematica, here it fools me....
> >
> > Regards Michael
> >
> > "dimitris" <dimmechan at yahoo.com> schrieb im Newsbeitrag
> > news:etqo3f$10i$1 at smc.vnet.net...
> > > Hello to all of you!
> > >
> > > Firstly, I apologize for the lengthy post!
> > > Secondly, this post has a close connection with a recent (and well
> > > active!)
> > > thread titled "Integrate" and one old post of mine which was based on
> > > a older
> > > post of David Cantrell. Since there was no response and I do consider
> > > the
> > > subject very fundamental I would like any kind of insight.
> > >
> > > In the section about Proper Integrals in his article, Adamchik
> > > mentions that the Newton-Leibniz formula (i.e. the Fundamental
> > > Theorem of Integral Calculus: Integrate[f[x],{x,a,b}]=F[b]-F[a],
> > > F[x]: an antiderivative), does not hold any longer if the
> > > antiderivative F(x) has singularities in the integration interval
> > > (a,b).
> > >
> > > To demonstrate this, he considers the integral of the function:
> > >
> > > f[x_] = (x^2 + 2*x + 4)/(x^4 - 7*x^2 + 2*x + 17);
> > >
> > > over the interval (0,4).
> > >
> > > Plot[f[x], {x, 0, 4}];
> > > (*plot to be displayed*)
> > >
> > > The integrand posseses no singularities on the interval (0,4).
> > >
> > > Here is the corresponding indefinite integral
> > >
> > > F[x_] = Simplify[Integrate[f[x], x]]
> > > ArcTan[(1 + x)/(4 - x^2)]
> > >
> > > Substituting limits of integration into F[x] yields an incorrect
> > > result
> > >
> > > Limit[F[x], x -> 4, Direction -> 1] - Limit[F[x], x -> 0, Direction -
> > >>1]
> > > N[%]
> > > NIntegrate[f[x], {x, 0, 4}]
> > >
> > > -ArcTan[1/4] - ArcTan[5/12]
> > > -0.6397697828266257
> > > 2.501822870767894
> > >
> > > This is because the antiderivative has a jump discontinuity at x=2
> > > (also at x = -2), so that the Fundamental theorem cannot be used.
> > >
> > > Indeed
> > >
> > > Limit[F[x], x -> 2, Direction -> #1]&/@{-1, 1}
> > > Show@Block[{$DisplayFunction=Identity},
> > > Plot[F[x],{x,#[[1]],#[[2]]}]&/@Partition[Range[0,4,2],2,1]];
> > > {-(Pi/2), Pi/2}
> > > (*plot to be displayed*)
> > >
> > > The right way of applying the Fundamental theorem is the following
> > >
> > > (Limit[F[x], x -> 4, Direction -> 1] - Limit[F[x], x -> 2, Direction -
> > >> -1]) +
> > > (Limit[F[x], x -> 2, Direction -> 1] - Limit[F[x], x -> 0, Direction
> > > -> -1])
> > > N[%]
> > > Pi - ArcTan[1/4] - ArcTan[5/12]
> > > 2.501822870763167
> > >
> > > Integrate works in precisely this way
> > >
> > > Integrate[f[x], {x, 0, 4}]
> > > N[%]
> > >
> > > Pi - ArcTan[1/4] - ArcTan[5/12]
> > > 2.501822870763167
> > >
> > > A little later, he (i.e. Adamchik) says "The origin of
> > > discontinuities
> > > along the path of integration is not in the method of indefinite
> > > integration but rather in the integrand."
> > >
> > > Adamchik mentions next that the four zeros of the integrand's
> > > denominator
> > > are two complex-conjugate pairs having real parts +/- 1.95334. It
> > > then
> > > seems that he is saying that, connecting these conjugate pairs by
> > > vertical line segments in the complex plane, we get two branch
> > > cuts...
> > >
> > > BUT didn't the relevant branch cuts for his int cross the real axis
> > > at x = +/- 2, rather than at x = +/- 1.95334?
> > >
> > > (NOTE: The difference between 1.95334 and 2 is not due to numerical
> > > error).
> > >
> > > Exactly what's going on here?
> > >
> > > Show[GraphicsArray[Block[{$DisplayFunction = Identity},
> > > (ContourPlot[#1[F[x + I*y]], {x, -4, 4}, {y, -4, 4}, Contours -> 50,
> > > PlotPoints -> 50, ContourShading -> False, Epilog -> {Blue,
> > > AbsoluteThickness[2], Line[{{0, 0}, {4, 0}}]},
> > > PlotLabel -> #1[HoldForm[F[x]]]] & ) /@ {Re, Im}]], ImageSize
> > > - > 500];
> > > (*contour plots to be displayed*)
> > >
> > >
> > > Consider next the following function
> > >
> > > f[x_] = 1/(5 + Cos[x]);
> > >
> > > Then
> > >
> > > Integrate[f[x], {x, 0, 4*Pi}]
> > > N[%]
> > > NIntegrate[f[x], {x, 0, 4*Pi}]
> > >
> > > Sqrt[2/3]*Pi
> > > 2.565099660323728
> > > 2.5650996603270704
> > >
> > > F[x_] = Integrate[f[x], x]
> > > ArcTan[Sqrt[2/3]*Tan[x/2]]/Sqrt[6]
> > >
> > > D[F[x], x]==f[x]//Simplify
> > > True
> > >
> > > Plot[f[x], {x, 0, 4*Pi}, Ticks -> {Range[0, 4*Pi, Pi/2], Automatic}]
> > > Plot[F[x], {x, 0, 4*Pi}, Ticks -> {Range[0, 4*Pi, Pi/2], Automatic}]
> > >
> > > The antiderivative has jump discontinuities at Pi and 3Pi inside the
> > > integration range
> > >
> > > Table[(Limit[F[x], x -> n*(Pi/2), Direction -> #1] & ) /@ {-1, 1}, {n,
> > > 0, 4}]
> > > {{0, 0}, {ArcTan[Sqrt[2/3]]/Sqrt[6], ArcTan[Sqrt[2/3]]/Sqrt[6]}, {-(P=
i/
> > > (2*Sqrt[6])), Pi/(2*Sqrt[6])},
> > > {-(ArcTan[Sqrt[2/3]]/Sqrt[6]), -(ArcTan[Sqrt[2/3]]/Sqrt[6])}, {0,
> > > 0}}
> > >
> > > Reduce[5 + Cos[x] == 0 && 0 <= Re[x] <= 4*Pi, x]
> > > {ToRules[%]} /. (x_ -> b_) :> x -> ComplexExpand[b]
> > > x /. %;
> > > ({Re[#1], Im[#1]} & ) /@ %;
> > > poi = Point /@ %;
> > >
> > > x == 2*Pi - ArcCos[-5] || x == 4*Pi - ArcCos[-5] || x == =
ArcCos[-5] ||
> > > x == 2*Pi + ArcCos[-5]
> > > {{x -> Pi - I*Log[5 - 2*Sqrt[6]]}, {x -> 3*Pi - I*Log[5 - 2*Sqrt[6]]},
> > > {x -> Pi + I*Log[5 - 2*Sqrt[6]]},
> > > {x -> 3*Pi + I*Log[5 - 2*Sqrt[6]]}}
> > >
> > > Of course F[4Pi]-F[0]=0 incorrectly.
> > >
> > > The reason for the discrepancy in the above result is not because of
> > > any problem with the fundamental theorem of calculus, of course; it is
> > > caused by the multivalued nature of the indefinite integral arctan.
> > >
> > >
> > > Show[GraphicsArray[Block[{$DisplayFunction = Identity},
> > > (ContourPlot[#1[F[x + I*y]], {x, 0, 4*Pi}, {y, -4, 4}, Contours ->
> > > 50, PlotPoints -> 50, ContourShading -> False,
> > > FrameTicks -> {Range[0, 4*Pi, Pi], Automatic, None, None},
> > > Epilog -> {{PointSize[0.03], Red, poi},
> > > {Blue, Line[{{0, 0}, {4*Pi, 0}}]}}] & ) /@ {Re, Im}]],
> > > ImageSize -> 500]
> > >
> > > So, in this example the discontinuities are indeed from the branch
> > > cuts that start and end from the simple poles of the integrand which
> > > is in agreement with V.A. paper!
> > >
> > >
> > > I think I am not aware of something fundamental!
> > > Can someone point out what I miss?
> > >
> > >
> > > Regards
> > > Dimitris
> > >
> > >
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