Iversen theorem

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If is an isolated essential singularity of an analytic function of a complex variable , then every exceptional value in the sense of E. Picard is an asymptotic value of at . For example, the values and are exceptional and asymptotic values of at the essential singularity . This result of F. Iversen [1] supplements the big Picard theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity.


[1] F. Iversen, "Récherches sur les fonctions inverses des fonctions méromorphes" , Helsinki (1914)
[2] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6


Iversen's theorem has been extended to subharmonic functions on , notably by W.K. Hayman, see [a1], [a2].


[a1] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)
[a2] W.K. Hayman, "Subharmonic functions" , 2 , Acad. Press (1989)
How to Cite This Entry:
Iversen theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article