case Schröder also illustrated with his method, , yielded , and hence . If is the action of a group element on a set, then the iterated function corresponds to a free group. Most functions do not have explicit general closed-form expressions for the n-th iterate. The table below lists some that do. Note that all these expressions are valid even for non-integer and negative n, as well as non-negative integer n. Note: these two special cases of are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table. Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref. Means of study Iterated functions can be studied with the Artin–Mazur zeta function and with transfer operators. In computer science In computer science, iterated functions occur as a special case of recursive functions, which in turn anchor the study of such broad topics as lambda calculus, or narrower ones, such as the denotational semantics of computer programs. Definitions in terms of iterated functions Two important functionals can be defined in terms of iterated functions. These are summation: and the equivalent product: Functional derivative The functional derivative of an iterated function is given by the recursive formula: Lie's data transport equation Iterated functions crop up in the series expansion of combined functions, such as . Given the iteration velocity, or beta function (physics), for the th iterate of the function , we have For example, for rigid advection, if , then . Consequently, , action by a plain shift operator. Conversely, one may specify given an arbitrary , through the generic Abel equation discussed above, where This is evident by noting that For continuous iteration index , then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group, The initial flow velocity suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the translation functional equation, See also Irrational rotation Iterated function system Iterative method Rotation number Sarkovskii's theorem Fractional calculus Recurrence relation Schröder's equation Functional square root Abel function Infinite compositions of analytic functions Flow (mathematics) Tetration Functional equation Notes References External Links Dynamical