### About some questions of differential algebra concerning to elementary functions

S. Zarzuela
1982 Publicacions matemàtiques
The ztudy o6 elementany 6unct .íon4, that .íz, o6 thoee 6unct.íon4 buílt up by u4 .íng xat.íonal 6unct .íonz oven Q, Exponentíalz, Logan.íthmz and Algebna.íc opetat.íon4, begán zomewhat z .íz tematt.cally w.íth the d.ívenze wohk4 that Jozeph L .íouv .ílle d.íd in the 1830'4 . Although h.íz capital a,ím waz to obta.ín tome tezul .t about the .íntegnat.íon by mean4 o6 elementany 6unet .íona, alongh.íz way he had to 4tudy azpect4 morse c.ítcunzctibed to the ztnuctune o6 these 6unet .íonz . About
more » ... he 6 .ínzt po .ínt we muzt zay tha .í, cetta.ínly, L.íouv.ílle obta.íned a te4ult that, even .ímptoved z .ínce that time, haz non changed e44entíally . It 4u6 .íce4, 6o~ezemple, to compare the wonk o6 Líouvílle g.íven in 121 and Rozenl.ícht'ein [6) . lde can obaenve then that the ínttoductíon o6 new language and new thecníque haz only línealízed the ptoblem, makíng clean whích pnopentíez o6 elementaty 6unetíon4 chatacteníze them . U4íng thí4 new language and new thecníque .ít íz po4zíble deal mote cleanly wíth 4ome queztíon4 about elementany 6unctíonz . Fon exemple, theít ítnedundancy, that waz alneady eztabl.í 4hed by Líouvílle hímzel6 in [21 and Handy in [11 . Now, the ínnedundancy íz a conzequence o6 an Strcuctune theonem (1 .2) that we can quíkly gíve by u4íng Ro4enlícht'z thecníquez . Alzo, the non zolvabílíty by'meanz o6 elementany 6unctíonz o6 centaín clazeícal tnanzcendental equatíon4 can be eazíly eztablízhed . (oe ztudy thíz in Sectíon 2 . Anothen tteatment o6 the Stnuctune theoter, and ínnedundance queztíonz (but non u4íng Rozenlícht'z thecníque4) can be 6ound in the Rízch'4 papen [5) .