Iversen theorem

2020 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)