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MathGroup Archive 2007

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Re: Help in analytical integration 3

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75202] Re: Help in analytical integration 3
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Fri, 20 Apr 2007 00:40:02 -0400 (EDT)
  • References: <f04nr2$8nu$1@smc.vnet.net>

Searching more the G-R tables of Integrals I could not find any
relevant formula.
So I should have been more helpful as regards my previous replies.

I tried various methods in order to get a closed form result by myself
but I
failed. So I consulted another forum and I posted your integral; in
order to make the things
easier I replaced specific values for {A1,B1,C1,D1}={1,2,3,4}

That is, I took the following integral

In[24]:=
Integrate[1/Sqrt[(1 - w^2)*(4 - w^2)*(9 - w^2)*(16 - w^2)], {w, 0, 1}]

Mathematica can't return a result. Trying now the substitution w^2=v.
So that

In[32]:=
(FullSimplify[#1, v > 0] & )[Dt[w]/Sqrt[(1 - w^2)*(4 - w^2)*(9 -
w^2)*(16 - w^2)] /. w -> Sqrt[v] /. Dt[v] -> 1]
Integrate[%, {v, 0, 1}]

Out[32]=
1/(2*Sqrt[(-16 + v)*(-9 + v)*(-4 + v)*(-1 + v)*v])

Out[33]=
Integrate[1/(2*Sqrt[(-16 + v)*(-9 + v)*(-4 + v)*(-1 + v)*v]), {v, 0,
1}]

Again Mathematica fails.

Note that also the other CAS I frequently used does not exhibit better
performance in this case.
Again the integrals are remained unevaluated.

Noting also on the timings spent by Mathematica for this integral I am
quite
convinced that it does not how to deal with them. For integrals like
the above one
(I think that) table look up is taken place. Mathematica through
pattern matching
searches for these integrals but it does not find anything relevant
and it returns quickly the
integral back.

In the other forum in which I consulted to, professor G. A. Edgar
replied me (and I have many
reasons to trust him!) that may be, there is no answer more elementary
than Schwarz-Christoffel
transformation.

See

http://planetmath.org/?op=getobj&from=objects&id=6289
http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping
http://mathworld.wolfram.com/ConformalMapping.html

In our case (after the change of the integration variable!) we have
to find the coordinate of the vertex of the appropriate
hexagon on the positive real axis.

Enough for now!
I hope you found my current more helpful (and more...hopeful!)
than previous ones.

I will be in touch for anything new!

Regards
Dimitris

PS

1)

Don't wait something simple. Quite the opposite!
For example the much more simpler definite integral

In[41]:=
Integrate[1/Sqrt[(1 - x^2)*(2 - x^2)], {x, 0, 1}]

returns

Out[41]=
EllipticK[1/2]/Sqrt[2]

whereas of course

In[42]:=
(Information[#1, LongForm -> False] & )[EllipticK]

>From In[42]:=
"EllipticK[m] gives the complete elliptic integral of the first kind
K(m)."

2)

Think also of contour integration. Take this advice as a further hint.
I don't try it by myself.

3)

Don't forget that if the things don't go well, you have NIntegrate in
your hands!
Symbolic results is better from numerical ones (especially working in
applied mathematics
field). But also numerical results is better than no...results at all!






=CF/=C7 ashesh =DD=E3=F1=E1=F8=E5:
> Hi all,
>
> I need to solve the following integration analytically:
>
> Integrate[1/Sqrt[(A1^2 - w^2)*(B1^2 - w^2)*(C1^2 - w^2)*(D1^2 - w^2)],
> {w, 0, A1}]
>
>  0 < A1 < B1 < C1 < D1
>
> Mathematica is unable to perform the same.
>
> Any help / tips in solving the same will be highly appreciated.
>
> Regards.



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