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Integration of Oscillating Function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg18445] Integration of Oscillating Function
*From*: Christian Honeker <xian at mpip-mainz.mpg.de>
*Date*: Wed, 7 Jul 1999 00:11:31 -0400
*Sender*: owner-wri-mathgroup at wolfram.com
Mathematica Users!
I am attempting to integrate the following function:
function[x_] :=
Exp[-(u^0.82*Cos[Pi*0.82]+0.00525*u/x)]*Sin[u^0.82*Sin[Pi*0.82]]
in the follow manner:
integratedfunction[x_] := (0.00525/(Pi*x^2)) * NIntegrate[function[x_],
{u,0,Infinity}, WorkingPrecision -> 50, MaxRecursion -> 18, MinRecursion
-> 3, AccuracyGoal -> 10]
where u is the integrand and x varies between 0.0001 and 1.
Mathematica responds with a complaint about the WorkPrecision criteria
not being met,
but still finds a solution.
The solution, however, depends strongly on the integration limits (i.e.
it is unstable). For example,
integration of function[0.0001] should result in zero. A plot of
function[0.0001]
vs. u makes this clear. Depending on the integration range, however,
Mathematica
finds values of integratedfunction[0.0001] between 0 and 65. The
solution for values
of x > 0.006 are correct, however. This is because the oscillation of
function[x_]
decreases rapidly as x increases.
I am hoping that by making Mathematica aware that the function
oscillates strongly,
it can then perform the integration correctly. Setting the option
Methods -> Oscillatory
does not seem to work.
Can anyone help me?
Background:
I am performing photon correlation spectroscopy (PCS) on complex fluids.
Function[x_]
(above) has its origin in the inverse Laplace transformation of the
Kaulrausch-Watts-Williams
(KWW) function. The integration needs to be performed in order to
determine the
distribution of relaxation times which correlate with the size of my
micelles.
I find that the size distribution (integratedfunction[x_]) has the
correct form
for x > 0.006 (see above). I would like to determine the other half of
the
size distribution (x < 0.006), however.
Thank you very much for any tips that you can provide!
Michael Bockstaller
Max Planck Institute for Polymer Research
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