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MathGroup Archive 2007

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symbolic integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74213] symbolic integration
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Wed, 14 Mar 2007 03:55:33 -0500 (EST)

I am reading about Symbolic Integration from Manuel Bronstein's
Symbolic Integration 1 and some references therein.

>From the articles:

"Indefinite and Definite Integration" by Kelly Roach (1992)
"The evaluation of Bessel functions via G-function identities" by
Victor Adamchik (1995)
"Definite Integration in Mathematica 3.0" by the same author
"Symbolic Definite Integration" by Daniel Lichtblau

I was able to figure out a lot of things on how Mathematica determines
indefinite and definite integrals.

What I don't understand (and I can't find any clear reference
anywhere) is how
Mathematica having evaluated an antiderivative "searches for" and
"figures out" possible singulaties in the integration range.

For example here (*non-integrable singularity at x=1/2*)

Block[{Message}, Integrate[1/(x - 1/2), {x, 0, 1}]]
Infinity

and here...

Integrate[1/Sqrt[x], {x, 0, 1}] (*integrable singularity*)
2

Also how about here

Integrate[1/(2 + Cos[x]), {x, 0, 2*Pi}]
(2*Pi)/Sqrt[3]

where in this example the integrand is a continuus function of x
but the indefinite integral return by Mathematica has a finite
discontinuity
at x=Pi

Integrate[1/(2 + Cos[x]), x]
Show@Block[{$DisplayFunction=Identity},Plot[%, {x,#[[1]],#[[2]]}]&/
@Partition[Range[0,2Pi,Pi],2,1]]

(2*ArcTan[Tan[x/2]/Sqrt[3]])/Sqrt[3]

Next

Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}]
Integrate::idiv : Integral of Log[Sin[x]^2]*Tan[x] does not converge
on \
{x,0,=CF=80}.
Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}]

Of course the integral is convergent.

Tr@(NIntegrate[Log[Sin[x]^2]*
          Tan[x],{x,#[[1]],#[[2]]},WorkingPrecision\[Rule]40]&/
@Partition[\
Range[0,Pi,Pi/2],2,1])
0``29.90647836759534

Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi/2, Pi}]
0

Plot[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}];
f = (FullSimplify[#1, 0 <= x <= Pi] & )
[Integrate[Log[Sin[x]^2]*Tan[x], x]]
(Plot3D[Evaluate[#1[f /. x -> x + I*y]], {x, 0.01, Pi}, {y, 0.01, Pi},
PlotPoints -> 50] & ) /@ {Re, Im};

Any insight will be greatly appreciate.



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