Re: Calculation of a not so simple integral
- To: mathgroup at smc.vnet.net
- Subject: [mg131225] Re: Calculation of a not so simple integral
- From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
- Date: Wed, 19 Jun 2013 01:26:29 -0400 (EDT)
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Hi, Roberto, You are right. Now, I tried your integral on my machine, and it gave a definite analytic result: Integrate[ Sin[x/2]^2/x^2/(x^2 - 4*Pi^2)^2/(x^2 + a^2)^2, {x, -Infinity, Infinity}, Assumptions -> a > 0] (3 E + 28 E \[Pi]^2 - 16 (-8 + E) \[Pi]^4 - 64 (-4 + E) \[Pi]^6)/(64 E (\[Pi] + 4 \[Pi]^3)^3) I do not like it, since it is independent of a. Also this: Integrate[ Sin[x/2]^2/x^2/(x^2 - 4*Pi^2)^2/(x^2 + a^2)^2, {x, -Infinity, Infinity}, Assumptions -> a > 0, PrincipalValue -> True] -(1/(64 E (\[Pi] + 4 \[Pi]^3)^3))(8 I E^2 \[Pi]^3 (1 + 4 \[Pi]^2) ExpIntegralEi[-1] - 8 \[Pi]^3 (15 \[Pi] + 28 \[Pi]^3 + I ExpIntegralEi[1] + 4 I \[Pi]^2 ExpIntegralEi[1]) + E (-3 - 28 \[Pi]^2 + 16 \[Pi]^4 + 64 \[Pi]^6 + 16 I \[Pi]^(5/2) MeijerG[{{0, 0, 1/2}, {}}, {{0}, {}}, -2 I, 1/2] + 64 I \[Pi]^(9/2) MeijerG[{{0, 0, 1/2}, {}}, {{0}, {}}, -2 I, 1/2])) Shows no dependence upon a. Further, this: tbl = Table[{a, NIntegrate[ Sin[x/2]^2/x^2/(x^2 - 4*Pi^2)^2/(x^2 + a^2)^2, {x, -Infinity, Infinity}]}, {a, 0.01, 3, 0.01}]; works and produces a table of values {a, int} that can be evaluated: ListPlot[tbl] and exhibits a dependence upon a. Alexei Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu -----Original Message----- From: Brambilla Roberto Luigi (RSE) [mailto:Roberto.Brambilla at rse-web.it] Sent: Monday, June 17, 2013 12:53 PM To: Alexei Boulbitch; mathgroup at smc.vnet.net Subject: Re: Calculation of a not so simple integral The integrand has NO poles on the real axis : in x=+/-(2*Pi) the integrand assume zero value : Limit[Sin[x/2]^2/(x^2 - 4 Pi^2), x -> 2 Pi] 0 Rob. -----Messaggio originale----- Da: Alexei Boulbitch [mailto:Alexei.Boulbitch at iee.lu] Inviato: luned=EC 17 giugno 2013 12.32 A: mathgroup at smc.vnet.net Oggetto: Re: Calculation of a not so simple integral (*Partial Fractions decomposition, Fourier Integrals The problem diagnostics: Calculate Integrate[ Sin[x/2]^2 / x^2 / (x^2-4*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 Hi, Roland, I checked your expression, it has a pole on the x axis. Check this Sin[x/2]^2 / x^2 / (x^2-4*Pi^2 )^2 / (x^2 + a^2)^2//StandardForm Could it be the case, that you have just written it down in a wrong way? Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu RSE SpA ha adottato il Modello Organizzativo ai sensi del D.Lgs.231/2001, i= n forza del quale l'assunzione di obbligazioni da parte della Societ=E0 avv= iene con firma di un procuratore, munito di idonei poteri. RSE adopts a Compliance Programme under the Italian Law (D.Lgs.231/2001). A= ccording to this RSE Compliance Programme, any commitment of RSE is taken b= y the signature of one Representative granted by a proper Power of Attorney= . Le informazioni contenute in questo messaggio di posta elettronica sono r= iservate e confidenziali e ne e' vietata la diffusione in qualsiasi modo o = forma. Qualora Lei non fosse la persona destinataria del presente messaggio= , La invitiamo a non diffonderlo e ad eliminarlo, dandone gentilmente comun= icazione al mittente. The information included in this e-mail and any attac= hments are confidential and may also be privileged. If you are not the corr= ect recipient, you are kindly requested to notify the sender immediately, t= o cancel it and not to disclose the contents to any other person.
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- Re: Calculation of a not so simple integral
- From: Alexei Boulbitch <Alexei.Boulbitch@iee.lu>
- Re: Calculation of a not so simple integral