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Defining constants causes integration to hang
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114215] Defining constants causes integration to hang
*From*: amannuc <amannuc at yahoo.com>
*Date*: Sat, 27 Nov 2010 03:37:37 -0500 (EST)
Dear Mathematica group,
I have found a situation where using the Constants option in Dt causes
an indefinite integration to hang. This is unexpected behavior because
when I don't provide the Constants information, the integration
finishes quickly. I am surprised that in this case providing more
information causes the problem. I am using Mathematica 7.0.
The following works:
n12[r_] := n1 - r dni/lD12 (* n1, dni and lD12 are actually constants
*)
g2[a_] := 2*a*Integrate[ Dt[n12[r], r] (n12[r]* Sqrt[n12[r]^2*r^2 -
a^2]),
{r, h2, h1}]
and the resulting definite integral seems correct but contains a
number of spurious derivatives such as: Dt[dni, r], Dt[lD12, r], etc.
These derivatives are zero because n1, dni and lD12 are actually
constants, not dependent on r. When I specify such in the Dt[n12[r],
r] expression, the integral never finishes executing. That is, in the
above expression I replace
Dt[n12[r], r] with
Dt[n12[r], r, Constants -> {n1, dni, lD12} ]. The integral now does
not finish.
So, if I do not use the Constants option in Dt, I have the spurious
derivatives in the calculated expression (e.g. Dt[dni,r]) which I
seek to remove. One way to do this is to use the transformation rule:
Dt[dni, r] -> 0 (for example). It turns out that I can apply this type
of rule one at a time, and once only.
For example, the following works fine:
g3[a] = g2[a] /. Dt[dni, r] -> 0
which removes the spurious derivatives of dni. If I try to remove
another spurious derivative, the integral hangs. Thus, the following
hangs (trying to remove Dt[dn1, r]:
g4[a] = g3[a] /. Dt[dn1, r] -> 0
It turns out the key fact is that I apply the transformation twice.
For example, I could switch the order of the above two transformation
statements. In that case, the first one completes but the second one
hangs.
Any help/advice is appreciated.
-Tony
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