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Re: Calculation of a not so simple integral
Posted:
Jun 21, 2013 5:45 AM


Am 15.06.2013 10:15, schrieb Roland Franzius: > (*Partial Fractions decomposition, Fourier Integrals > > The problem diagnostics: > > Calculate > > Integrate[ Sin[x/2]^2 / x^2 / (x^24*Pi^2 )^2 / (x^2 + a^2)^2 , > {x,oo,oo}, Assumptions> a>0] > > The integrand is nonnegative, has no poles on the real line and decays > rapidly ~ x^10 as x>+oo > > Mathematica version 6+8 are failing to give a result in a reasonable > time of calculation. > Time constraint does not seem to work > > The indefinite integral is calculated readily with an obscure complex > result, the primitive function giving completely useless limits for > x>0,+oo > > The FourierTransform with Limit k>0 seems to give a correct result > compared to numerical integration for values of > a > 1/20 > > *)
Remains to be noted that suddenly today in the morning my Mathematica 8 (local install with license server) seems to have been updated over the internet with respect to this integral.
While yesterday the result was MeiersG, when I came back after 2 hours, suddenly we have amuch faster and slightly better result
In[14]:= ires=Timing[ Integrate[Evaluate[Sin[x/2]^2 f[x,2\[Pi],a]],{x,\[Infinity],\[Infinity]},Assumptions>0<a<1]]
Out[14]= {64.865, ( 3 a^7 + 28 a^5 \[Pi]^2 + 16 a^2 (7 + 6 a) \[Pi]^4 + 64 (3 + 2 a) \[Pi]^6)/(64 a^5 \[Pi]^3 (a^2 + 4 \[Pi]^2)^3)}
still wrong, the exponential terms are missing.
But at least for a>1 we get the algebraically correct result.
In[31]:= iresg=Timing[ Integrate[Evaluate[Sin[x/2]^2 f[x,2\[Pi],a]],{x,\[Infinity],\[Infinity]},Assumptions>1<a<\[Infinity]]]
Out[31]= {63.477, ( E^a (16 a^2 (7 + a) \[Pi]^4 + 64 (3 + a) \[Pi]^6 + E^a (3 a^7 + 28 a^5 \[Pi]^2 + 16 a^2 (7 + 6 a) \[Pi]^4 + 64 (3 + 2 a) \[Pi]^6)))/(64 a^5 (a^2 \[Pi] + 4 \[Pi]^3)^3)}
With Assumptions > a>0, Integrate produces both results as a conditional expression. It Contains the difference of two imaginary Log's.
So I conclude there seem to exists problems with the partial fractions decomposition engine and in the simplification of complex Log's and Atan's.
Of course this is a nontrivial matter because it involves complex contour integrals which cannot be desribed by the endpoints only.

Roland Franzius



