Commentatio in fractionem continuam, qua illustris La Grange potestates binomiales expressit

TL;DR

This paper explores Euler's continued fraction representations of functions like logarithm and arctangent, including their derivation, evaluation at complex points, and connections to binomial powers, providing insights into classical analysis techniques.

Contribution

It presents a translation and analysis of Euler's original work on continued fractions for various functions, highlighting novel derivations and evaluations.

Findings

Euler's continued fractions for log and arctan are derived and analyzed.

Evaluation at complex points reveals new identities and representations.

Connections between binomial powers and continued fractions are established.

Abstract

Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for "vanishing" n, and for infinite n and deduces a continued fraction for log, arctan, etc. The paper is translated from Euler's Latin original into German.