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Iversen theorem

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Iversen theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Iversen_theorem&oldid=27313
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article