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
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along which
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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) |
Asymptotic value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_value&oldid=18288