der auf die ate Wurzel yon a fiihr~. Dieses Beispiel diirfte ein gewisses Interesse darbieten. Denn die iterirte Function ist sehr eiufach rational und die Convergenz ist sehr stark. Ausserdem wird die Variable nicht nur auf reale We~he beschriinkL Ftir den Fall der Quadratwurzel (n ~ 2) ist das Ergebnlss tiberraschend einfach. Dann zerf~llt die Ebene der complexen Zahlen in zwei Convergenzbereiche t die durch eine gerade Linie yon einander getrennt sin& Diese gerade Linie enthNt alle diejenigen x, welche gleichweit yon den beiden Punkten, die die Quadratwurzeln aus a darstellen, en~fern~ sind; w isi also die Mittelsenkrechte der Verbindungslinie der beiden Punkto.*) Lieg~ x auf dieser Linie, so convergirt der Algorithmus nieht. In jedem andern Falle convergirt er gegen denjenigen Werth yon ~/a'~ der dem Anfangswerthe x am niichsten liegt. Der Grenzwerth l~isst sich auch als eine unendliche l~eihe yon rafionalen Functionen yon x darstellen, n,imlich in der Form x + (xi-x) + (x~-xl) + .... *) Ich will znich im Folgenden immer der bequemen und anschaulichen aus der geometrischen Darstellung der complexen Zahlen en~springenden Ausdrucksweise bedienen.
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