Definite Integration in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg74428] Definite Integration in Mathematica
- From: "dimitris" <dimmechan at yahoo.com>
- Date: Wed, 21 Mar 2007 02:46:53 -0500 (EST)
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]}, {-(Pi/
(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