Iversen theorem
2010 Mathematics Subject Classification: Primary: 32H25 [MSN][ZBL]
If $a$ is an isolated essential singularity of an analytic function $f(z)$ of a complex variable $z$, then every exceptional value $\alpha$ in the sense of E. Picard is an asymptotic value of $f(z)$ at $a$. For example, the values $\alpha_1=0$ and $\alpha_2=\infty$ are exceptional and asymptotic values of $f(z) = \mathrm{e}^z$ at the essential singularity $a=\infty$. This result of F. Iversen [Iv] supplements the big Picard theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity.
Iversen's theorem has been extended to subharmonic functions on $\R^n$, notably by W.K. Hayman, see [HaKe], [Ha].
References
[CoLo] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets", Cambridge Univ. Press (1966) pp. Chapt. 1;6 |
[Ha] | W.K. Hayman, "Subharmonic functions", 2, Acad. Press (1989) |
[HaKe] | W.K. Hayman, P.B. Kennedy, "Subharmonic functions", 1, Acad. Press (1976) |
[Iv] | F. Iversen, "Récherches sur les fonctions inverses des fonctions méromorphes", Helsinki (1914) |
Iversen theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Iversen_theorem&oldid=27313