» Ritt J.F. - Integration in finite terms, Liouville theory of elementary methods
Ritt J.F. - Integration in finite terms, Liouville theory of elementary methods
||Integration in finite terms, Liouville theory of elementary methods
||During the period between 1&33 &"<* 1S41» J« Liouville presented a theory of integration in finite terns. He determined the form which the integral of an algebraic function must have when the integral can Tie expressed with the operations of elementary mthe-inatical analysis, carried out a finite number of tines. He showed that the elliptic integrals of the first and second kinds have no elementary expressions. He proved that certain sinple differential equations cannot be solved by elementary procedures, bis papers contain other remarkable applications of his theory.
The questions treated by Liouville are questions which occur to every strong undergraduate student of natbemtics. Nevertheless Liouville's work never received very wide attention. It has always been something which everyone would like very much to know about hut which very few undertake to study.
During the nineteenth century, extremely little was done in direct continuation of Liouville's work.* About forty years ago, the Russian BBthenaticiaa Mordukhai-Boltcvskoi began to write on Liouville's theory and contributed extensively to it. In particular, he published a book on the integration of transcendental functions and one on the integration in finite terns of Linear differential liquations. Through his influence, the subject seens to have been more widely studied in Russia than elsewhere. The present writer published sore work on these questions between 1<#3 and 19"?• At the present tine, Ostrowski is writing oa the subject.
This monograph jives an account of Liouville's work and of seme of that cf his few followers. Oa the basis of what has already been said, a glance through the chapters, or even over the table of contents, will give a sufficient idea of the topics covered.
I should like, however, to say sonething in regard to the treatment given here of Liouville's work. Liouville's methods are ingenious and beautiful. From the formal standpoint, they are entirely sound. There are, however, certain questions connected with the
• Та be sure, (here appeared the Pieard-Vessiot theory of lineiг differencial equations, which furnishes, for such equation», result* analogous Co ehose of (aloii far aJtebnic equation». Recent mark 01" E. R. KoJehin ha* ЬгофЬс rigor »nd sinplicicy to the Picard-Vessiot theory.
Che should perhaps mention also the remarkable work of Bran» on the algebraic solutions of the equations of celestial mechanics. (Acta Hathwumtica. Vol. XI.)