Asymptotic value

A limit value along some path. More exactly, a complex number $\alpha$ or $\alpha=\infty$ is called an asymptotic value for a function $f(z)$ of the complex variable $z$ at a point $a$ of the closure $\overline D$ of its domain of definition $D$ if there exists a path $L$: $z=z(t)$, $0\leq t<1$, $L\subset D$, terminating at $a$, i.e. so that

\[ \lim_{t → 1 - 0} z(t) = a \]

along which

$$\lim_{z\to a}f(z)=\alpha,\quad z\in L.$$

For instance, at the point $a=\infty$ the function $f(z)=e^z$ has the asymptotic values $\alpha_1=0$ and $\alpha_2=\infty$ along the paths $L_1$: $z=-t$, $0\leq t<+\infty$, and $L_2$: $z=t$, $0\leq t<+\infty$, respectively. Sets of asymptotic values play an important role in the theory of limit sets (cf. Limit set).

If $f(z)$ has two different asymptotic values at $a$, $a$ is called a point of indeterminacy for the function $f(z)$. For any function $f(z)$, 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 $L$ is a set $E\subset\partial D$ rather than a single point $a\in\partial D$, one also speaks of the asymptotic value $A(f,E)$ associated with $E$.

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 $f(z)$ be an entire function with $n$ distinct (finite) asymptotic values at the point $\infty$. Then $f(z)$ must be of order $\geq n/2$. 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)