PR???FACE. "La premi???re Partie de cet Ouvrage est consacr???e ??? l'exposition d'une th???orie des fonctions, d'apr???s les id???es de Cauchy. Le principe fondamental de cette th???orie est la consid???ration des fonctions d'une variable imaginaire. Il appara???t pour la premi???re fois dans le M???moire c???l???bre de 1825 sur les int???grales d???finies prises entre des limites imaginaires. Depuis, par les travaux de Cauchy et des g???om???tres qui ont suivi ses traces, il a re???u des d???veloppements tels, et a ...
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PR???FACE. "La premi???re Partie de cet Ouvrage est consacr???e ??? l'exposition d'une th???orie des fonctions, d'apr???s les id???es de Cauchy. Le principe fondamental de cette th???orie est la consid???ration des fonctions d'une variable imaginaire. Il appara???t pour la premi???re fois dans le M???moire c???l???bre de 1825 sur les int???grales d???finies prises entre des limites imaginaires. Depuis, par les travaux de Cauchy et des g???om???tres qui ont suivi ses traces, il a re???u des d???veloppements tels, et a conduit ??? la d???couverte d'un si grand nombre de v???rit???s nouvelles, que son importance est aujourd'hui universellement reconnue. Cependant on constate avec regret que, dans quelques ouvrages consacr???s ??? cet ordre de recherches, on ne rend pas ??? Cauchy la justice qui lui est due. Dans la th???orie de Cauchy, la marche de la variable imaginaire est figur???e par le mouvement d'un point sur un plan. Pour repr???senter les fonctions qui acqui???rent plusieurs valeurs pour une m???me valeur de la variable, Riemann regardait le plan comme form??? de plusieurs feuillets superpos???s et r???unis par des soudures, de mani???re que la variable puisse passer d'un feuillet ??? un autre en traversant une ligne de raccordement. La conception des surfaces ??? feuillets multiples pr???sente quelques difficult???s; malgr??? les beaux r???sultats auxquels Riemann est arriv??? par cette m???thode, elle ne nous a paru pr???senter aucun avantage pour l'objet que nous avions en vue. L'id???e de Cauchy se pr???te tr???s- bien ??? la repr???sentation des fonctions multiples; il suffit de joindre ??? la valeur de la variable la valeur correspondante de la fonction, et, quand la variable a d???crit une courbe ferm???e et que la valeur de la fonction a chang???, d'indiquer ce changement par un indice."
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