Specially Challenging Integrals ??
monir
1) I've 3 definite integrals I'm trying to evaluate analytically; i.e.
to find the EXACT evaluation expressions.
The independent variable is "X", and the constants are: a, b, c, d, e,
f, g, h, j, k, m, n, H, B, C, G, and F.
They all have the same denominator D, but each has a different
numerator.
...Int1 = int[X=0 to inf] { a + b*COS(X) + c*SIN(X) } / D^(3/2) dX
...Int2 = int[X=0 to inf] { d*COS(X) + e*SIN(X) + f*X*COS(X) + g*X*SIN
(X) } / D^(3/2) dX
...Int3 = int[X=0 to inf] { m + h*SIN(X) + j*COS(X) + k*X*COS(X) +
n*X*SIN(X) } / D^(3/2) dX
...D = { H + B*X + C*X^2 + F*COS(X) + G*SIN(X) }
If D = 0.0, the integrals don't exist.
The numerical values of all the constants are known.
2) I've checked advanced Tables of Integrals, special functions and
many texts for guidance on the Exact Analytical Expressions. Couldn't
find anything that even closely resemble the integrals in question.
This by no means suggests that the exact expressions don't exist, I
just simply couldn't find them.
3) I've tried many transformations, change of variables, integration
by parts (more than once), substitutions, etc. Absolutely dead-end!
In fact in most cases, I ended up with "horrible" looking integrals!!
and in one attempt after considerable work and many hours of analysis
I ended up with Int1 = Int1 "unintended circular transformation"!
There is a good change that some mathematicians might have directly or
indirectly come across such troublesome integrals through their own
research or experience, and would be kind enough to advise or provide
some guidance on how-to.
Your help would be greatly appreciated.
Kind regards.
Monir
[mod note: you seem to want to resist what looks to me like the most
plausible conclusion, namely "the definite integral does not exist"
(in the sense that it can't be written down in terms of functions that are
deemed "standard"). Also note the web page
http://integrals.wolfram.com/index.jsp
which sometimes is useful for tricky integrals.]
G. A. Edgar
I suggest, instead of the monsters you listed, ask for "closed form" of
integral(0 to infinity) 1/(x^2+cos(x))^(3/2) = 0.25443247203094
I still think it is hopeless. But maybe a challenge someone will take
up.
In article <hc4onn$hdg$1...@crackerjack.ma.ic.ac.uk>, monir
<mon...@mondenet.com> wrote:
[...]
>
> ...Int1 = int[X=0 to inf] { a + b*COS(X) + c*SIN(X) } / D^(3/2) dX
[...]
>
> ...D = { H + B*X + C*X^2 + F*COS(X) + G*SIN(X) }
>
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
stefano.c...@googlemail.com
otherwise either 1) the problem does not have a solution or 2) the
problem is to difficult for humans.
[mod note: the web page I mentioned earlier in this thread
is I think just using mathematica, although I think it will time out
after a certain time if it hasn't found an answer. Another thing
the OP can try is to try some explicit easy examples (when most
but not all of the extra variables are zero, and the rest are 1),
like Prof Edgar suggested, and then integrate numerically and then try
to see if it can recognise the answers that come up. But again let
me stress that the integrals might already be in their "simplest form"...]
monir
<stefano.cardanob...@googlemail.com> wrote:
> Try Mathematica. If it has a solution, Mathematica will find it;
> [mod note: the web page I mentioned earlier in this thread
Thank you all for your replies.
1) The 3 "monster" integrals I listed in the OP are in their "simplest
form", and although they share common elements I thought it would be
helpful to list them in full so that any of them might trigger a
particular thought.
2) The difficulty in analytically evaluating the quoted integrals
appears to be associated partly with the exponent (3/2) but mostly
with the combination of the quadratic trinomial a+b*x+c*x^2 and the
trigonometric functions in D.
I thought initially it would be the other way around!
3) Considering the 1st definite integral:
...Int1 = int[X=0 to inf]{ a + b*Cos[X] + c*Sin[X]}/D^(3/2) dX
...D = { H + B*X + C*X^2 + F*Cos[X] + G*Sin[X] }
I eliminated most terms in Int1 above to reproduce Dr. Edgar's
example:
...I4 = integrate { a /( C*X^2 + F*Cos[X] )^(3/2) dX
and tried it as indefinite integral on the Wolfram Mathematica OnLine
Integrator.
The integrator was unable to finish in the "time allotted".
My goal was to examine the terms of the solution formula provided by
Mathematica and possibly detect how the integral was broken down to
its manageable parts (in the absence of Mathematica's steps).
Appreciating that the simplified denominator in the above example has
only X^2 and Cos[X] terms, I thought it would have been a reasonable
starting point.
What is noticeable is that Mathematica OnLine Integrator didn't say "a
closed form formula for the integral (I4) doesn't exist". It simply
says "time expired" and that's why it "didn't finish". Perhaps I'm
reading too much into it!
(I don't have access to a full blown Mathematica to re-try with no
time limit.)
4) The following sample "simplified" INDEFINITE integrals have closed
form solution:
(using my OP constants notations)
...I5 = integrate { a /( A + B*X + C*X^2 )^(3/2) dX
...I6 = integrate { a /( H + F*Cos[X] + G*Sin[X] )^(3/2) dX
(~ 5-page long formula)
...I7 = integrate { (b*Cos[X] + c*Sin[X] ) / ( H + F*COS(X) + G*SIN
(X) )^(3/2) dX
(~ 10-page long formula)
Adding Sin[X] and/or Cos[X] term to the 2nd deg poly in I5 above, or
adding "X" and/or "X^2" term in I6 or I7, make the relevant integral
untouchable, at least so far in my search!
5) I tried earlier to use the sine and cosine power series expansions
(in D for Int1, Item 3 above):
...Sin[X] = X + Sum[k=1 to inf] (-1)^k X^(2k+1) / (2k+1)!
...Cos[X] = 1 - 0.5*X^2 + Sum[k=2 to inf] (-1)^k X^(2k) / (2k)!
and combining the first few terms D becomes:
...D = {p + q*X + s*X^2 + F*Sum[k=2 to inf] (-1)^k X^(2k)/(2k)! +
G*Sum[k=1 to inf] (-1)^k X^(2k+1)/(2k+1)! }
and the integral:
...Int1 = int[X=0 to inf] {a + b*Cos(X) + c*Sin(X)} / D^(3/2) dX
Couldn't proceed any further!!
6) Another thought I had was whether in addition to searching advanced
"Tables of Integrals" one perhaps should also search advanced "Tables
of Derivatives" !! I don't think such Tables exist, simply because
derivatives follow specific rules and they're much easier to do than
integrals.
But, could the derivative of a particular formula or special function
(s) have the form that closely resembles the integral in question ??
Will keep trying, and would very much appreciate any comments/
suggestions you experts might have.
Regards.
Monir
Robert Israel
There is an algorithm, the Risch algorithm, to determine whether an elementary
function has an elementary antiderivative. In principle that
could be applied to your integral. AFAIK the mixed algebraic-transcendental
case (which is what this would need) has not been completely implemented in
any of the major Computer Algebra Systems. I would guess, however, that your
integrals do not have elementary antiderivatives except in trivial cases
(e.g. when the numerator of the integrand is 0).
On the other hand, definite integrals (especially improper ones) can often
have closed forms even though the corresponding antiderivative does not.
For example, this can sometimes be done by complex variable methods (residues
etc). As far as I know, there is no general theory that can decide whether or
not a definite integral can be expressed in any sort of closed form.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
monir
<isr...@math.MyUniversitysInitials.ca> wrote:
> monir <mon...@mondenet.com> writes:
> > On Oct 27, 9:07�am, "stefano.cardanob...@googlemail.com"
Thank you for suggesting the Risch algorithm.
I've just sent PM to Prof. K. Geddes, Univ. of Waterloo, asking for
advice. I've been told he is an expert in Risch and integrals of
mixed algebraic-transcendental functions.
Will report any progress.
Regards.
Monir