Title The computational power of iterating analytic functions Author Warren D. Smith Abstract We describe a meromorphic function $g(z)$ and an entire analytic function $h(z)$ whose iterations each simulate a universal computer. The simulation is unaffected by ``noise'' of bounded magnitude. Consequently, numerous problems about such iterations are undecidable. There are senses in which $g(z)$ and $h(z)$ are simplest possible: we show iterating rational functions cannot cause Turing universal behavior. Finally, we consider Blum, Cucker, Shub, and Smale's notion of ``undecidable'' problems for a machine capable of performing real arithmetic operations. B,C,S\&S had claimed without proof that membership of a point in the ``Mandelbrot set'' $M$ was undecidable in their sense. Although we provide a proof, we argue that this statement is very deceptive. For example, we show (assuming Douady's conjecture that $M$ is ``locally connected'') that this problem {\em is} decidable if the point is known to be either in the interior or complement of $M$, or if the point is known to have algebraic coordinates. Neither the inscrutable nature of $M$, nor the fact that its boundary has Haussdorf dimension $2$, need have anything to do with the undecidability. To see that, we show the well behaved set $S_e = \{x,y | y \le e^x \}$ is BCSS-undecidable. Again, though, we have decidability if $(x,y)$ is known to lie in the interior or complement of $S_e$, or if $x$ and $y$ are known to be algebraic numbers. For these reasons we recommend a redefinition of BCSS-undecidability. Keywords Elliptic functions, meromorphic functions, simple computers, counter machines, Turing universal, entire functions, iteration of rational functions, repetition count functions, Mandelbrot set, Julia set. Not patentable.