 A simpler version of the argument yields two useful continued fraction expansions of the exponential function.From
It is easily shown that the Taylor series expansion of arctan "z" in a neighborhood of zero is given by

The continued fraction of Gauss can be applied to this identity, yielding the expansion

which converges to the principal branch of the inverse tangent function on the cut complex plane, with the cut extending along the imaginary axis from "i" to the point at infinity, and from −"i" to the point at infinity.This particular continued fraction converges fairly quickly when "z" = 1, giving the value π/4 to seven decimal places by the ninth convergent.