Very cute! But taking it more seriously than it was probably intended: In general (thinking along the same lines as Marc) suppose that f(x; a) is a function f( ; a) : R -> R depending analytically on x, for a in some open set of R^2. Then we want to solve (*) x = f(x;a) Then: 1) For which (x,a) will iterating f( ;a) on the RHS of (*) converge to a fixed point x=c ? and 2) When will the fixed point be a uniquely determined analytic function c(a) of a ? In case anyone cares: To take a well-known case, if we set f(x; a) := a^x (a > 0), then the behavior is pretty intricate: Iterating the RHS of (*) converges precisely for a in the interval [e^(-e), e^(1/e)], but not always for all real x. This will occur for all real x if a <= 1. But if a lies in (1, e^1/e) then f(x) = a^x has two real fixed points x_0 < x_1 in the interval (1,e), and iteration of the RHS of (*) will converge (to x_1) precisely for x <= x_1. Finally, for a = e^(1/e), the expected limiting behavior occurs: f(x) = e^(x/e) has just one fixed point at x = e, and iteration of the RHS of (*) converges precisely for x <= e. And things get even more hairy if a is allowed to be complex, with f(x;a) most conveniently viewed instead as g(z;c) := exp(c*z) (so c is a fixed choice of log(a)). See "Iteration of the exponential function" at < http://eretrandre.org/rb/files/Wright1947_217.pdf >. --Dan Jim wrote: << . . . "Problem: Solve x = ax + b for x. Solution: x = a(ax+b) + b = a^2 x + ab + b = a(a(ax+b)+b)+b= a^3 x + a^2 b + ab + b ... = (assuming |a| < 1) lim_{n \rightarrow \infty} a^n x + b sum_{i=0}^{\infty} a^i = 0 + b/(1-a). This also holds by analytic continuation for all a neq 1." . . .