R: Re: Difficult antiderivative

*To*: mathgroup at smc.vnet.net*Subject*: [mg128842] R: Re: Difficult antiderivative*From*: "Brambilla Roberto Luigi (RSE)" <Roberto.Brambilla at rse-web.it>*Date*: Fri, 30 Nov 2012 05:54:45 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20121129110606.EA8C968CD@smc.vnet.net>

Many thanks Alexei! You comments is very illuminating! I've also found that the more general integral Integrate[ (x^m)ArcCosh[a/x]/Sqrt[r^2-x^2],x] "exist" (by means of elementary functions or usual special functions Used by Mathematica) for m=1,3,5... but nor for m=0,2,4... Another antiderivative (to which the previous can be reduced)is Integrate[ (x^m)ArcSin[k/x]/Sqrt[1-x^2],x] In this case Mathematica solves it for k =|= 1 only if m is odd but for k=1 Mathematica solves also for m even. I'm asking if there exist any general criterion (at least for simple combinations of elementary functions, as in my examples) that tell us about the existence of antiderivative in the field of a set of chosen elementary functions. Can I add to this set other less elementary functions (like Pailev=E9 trascendentans) in order to catch the missing antiderivative? Cheers, Rob -----Messaggio originale----- Da: Alexei Boulbitch [mailto:Alexei.Boulbitch at iee.lu] Inviato: gioved=EC 29 novembre 2012 12.06 A: mathgroup at smc.vnet.net Oggetto: Re: Difficult antiderivative Q.: may be that the antiderivative does not exist? The numerical integral (b<r<a) NIntegrate[ ArcCosh[a/x]/Sqrt[r^2-x^2],{x,b,r}] Don't give any problem. Any help very appreciated (and considered). Cheers, Rob Hi, Rob, Yes, it is a general case that indefinite integrals cannot be expressed in terms of some finite combination of analytical and special functions. That is a more mathematically correct expression of the thing you obviously have in mind when writing "does not exist". Even more, most of indefinite integrals have this property, and only smaller part of them can be expressed in terms of analytical and special functions. It is also common that the indefinite integral "does not exist" (using your expression), while the definite one does. Have fun, 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<mailto:alexei.boulbitch at iee.lu> RSE SpA ha adottato il Modello Organizzativo ai sensi del D.Lgs.231/2001, in forza del quale l'assunzione di obbligazioni da parte della Societ=E0 avviene con firma di un procuratore, munito di idonei poteri. RSE adopts a Compliance Programme under the Italian Law (D.Lgs.231/2001). According to this RSE Compliance Programme, any commitment of RSE is taken by the signature of one Representative granted by a proper Power of Attorney. Le informazioni contenute in questo messaggio di posta elettronica sono riservate e confidenziali e ne e' vietata la diffusione in qualsiasi modo o forma. Qualora Lei non fosse la persona destinataria del presente messaggio, Lainvitiamo a non diffonderlo e ad eliminarlo, dandone gentilmente comunicazione al mittente. The information included in this e-mail and any attachments are confidential and may also be privileged. If you are not the correct recipient, you are kindly requested to notify the sender immediately, to cancel it and not to disclose the contents to any other person.

**References**:**Re: Difficult antiderivative***From:*Alexei Boulbitch <Alexei.Boulbitch@iee.lu>