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MathGroup Archive 2004

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Interesting integral problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47716] Interesting integral problem
  • From: Arturas Acus <acus at itpa.lt>
  • Date: Fri, 23 Apr 2004 02:30:44 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear Group,

The following integral http://www.itpa.lt/mathematica/IntegralProblem.nb
was my headache for a week, still I cannot decide if it can be expressed
using elementary or special functions. The problem is if one has two
quite similar integrals, one of them can be calculated explicitly, other
seems not. What is so special about the first? If somebody could look at
it I would be very gratefull.

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Cell["\<\
Dear Group,

The following integral was my headache for a week, still I cannot \
decide if it can be expressed using elementary or special functions. \
If somebody could look at it I would be very gratefull.

 The task is to integrate  integral bellow in spherical coordinates \
(do not forget to multiply it by Sin[\[Theta]]).\
\>", "Text",
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If we first integrate by \[Theta] (integratation of \
\[CurlyPhi] first gives less promising rezult and looks like a dead \
end) , after a lot of simplification hints we can arrive to the \
following integral over \[CurlyPhi]:\
\>", "Text",
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with a=Cos[4 \[CurlyPhi]]. The following numerical check  \
ensures, that no mistake was made.\
\>", "Text",
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The problematic part is with the Log[ ]. Can it be \
integrated?\
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If not, then what so special (from algebraic point of view) \
has the following quite similar integral, which do can be integrated \
explicitly (even undefined integration actually can be \
performed)\
\>", "Text",
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First integrating by \[Theta] and simplifying we get\
\>", \
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is of the same complexity as previous integral, but can be \
explicitly integrated:\
\>", "Text",
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However if we alternatively rewrite the problematic part, \
the integration fails with $RecursionLimit and $IterationLimit \
errors. It also fails after Apart[], etc... etc....\
\>", "Text",
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I rewrited the integral using various substitutions, in \
many forms. All in vain. Thought both these integrals seems quite \
similar, one can be integrated, whereas other no. This looks quite \
interesting and somehow unusual for me. For example if you look in \
Grandstein Rhyzik integral tables, you will find quite general \
formulas, integrability of which do not depend on, say what \
polynomial coefficients are. I home that somebody can shield a light \
on my problem.   THANKS.\
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