Asymptotic value

A limit value along some path. More exactly, a complex number or is called an asymptotic value for a function of the complex variable at a point of the closure of its domain of definition if there exists a path : , , , terminating at , i.e. so that

along which

For instance, at the point the function has the asymptotic values and along the paths : , , and : , , respectively. Sets of asymptotic values play an important role in the theory of limit sets (cf. Limit set).

If has two different asymptotic values at , is called a point of indeterminacy for the function . For any function , defined in a simply-connected plane domain, the set of points of indeterminacy is at most countable.

The above definition of asymptotic value refers to asymptotic point values. If the limit set of a curve is a set rather than a single point , one also speaks of the asymptotic value associated with .

References

[1]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[2]	G.R. MacLane, "Asymptotic values of holomorphic functions" , Rice Univ. Studies, Math. Monographs , 49 : 1 , Rice Univ. , Houston (1963)

Comments

The most famous results on asymptotic values is the Denjoy–Carleman–Ahlfors theorem. Let be an entire function with distinct (finite) asymptotic values at the point . Then must be of order . This result was conjectured by A. Denjoy (1907). The first complete proof was given by L. Ahlfors (1929), after T. Carleman had obtained a less sharp result. See, for example, [a1], Sect. 60.

References

[a1]	A. Dinghas, "Vorlesungen über Funktionentheorie" , Springer (1961)