equation of the functional roots of the identity map. For example, for and , both and are solutions; so the expression doesn't denote a unique function, just as numbers have multiple algebraic roots. The issue is quite similar to the expression "0/0" in arithmetic. A trivial root of f can always be obtained if fs domain can be extended sufficiently, cf. picture. The roots chosen are normally the ones belonging to the orbit under study. Fractional iteration of a function can be defined: for instance, a half iterate of a function is a function such that . This function can be written using the index notation as . Similarly, is the function defined such that , while may be defined as equal to , and so forth, all based on the principle, mentioned earlier, that . This idea can be generalized so that the iteration count becomes a continuous parameter, a sort of continuous "time" of a continuous orbit. In such cases, one refers to the system as a flow. (cf. Section on conjugacy below.) Negative iterates correspond to function inverses and their compositions. For example, is the normal inverse of , while is the inverse composed with itself, i.e. . Fractional negative iterates are defined analogously to fractional positive ones; for example, is defined such that , or, equivalently, such that . Some formulas for fractional iteration One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows. First determine a fixed point for the function such that . Define for all n belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates. Expand around the fixed point a as a Taylor series, Expand out Substitute in for , for any k, Make use of the geometric progression to simplify terms, There is a special case when , This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on Conjugacy. Example 1 For example, setting gives the fixed point , so the above formula terminates to just which is trivial to check. Example 2 Find the value of where this is done n times (and possibly the interpolated values when n is not an integer). We have . A fixed point is . So