Difference between revisions of "Iversen theorem"
From Encyclopedia of Mathematics
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| − | + | If $a$ is an isolated essential singularity of an analytic function $f(z)$ of a complex variable $z$, then every [[Exceptional value|exceptional value]] $\alpha$ in the sense of E. Picard is an [[Asymptotic value|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 {{Cite|Iv}} supplements the big [[Picard theorem]] on the behaviour of an analytic function in a neighbourhood of an essential singularity. | |
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| + | Iversen's theorem has been extended to subharmonic functions on $\R^n$, notably by W.K. Hayman, see {{Cite|HaKe}}, {{Cite|Ha}}. | ||
| − | + | ====References==== | |
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| − | + | |valign="top"|{{Ref|CoLo}}||valign="top"| E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets", Cambridge Univ. Press (1966) pp. Chapt. 1;6 | |
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| − | + | |valign="top"|{{Ref|Ha}}||valign="top"| W.K. Hayman, "Subharmonic functions", '''2''', Acad. Press (1989) | |
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| + | |valign="top"|{{Ref|HaKe}}||valign="top"| W.K. Hayman, P.B. Kennedy, "Subharmonic functions", '''1''', Acad. Press (1976) | ||
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| + | |valign="top"|{{Ref|Iv}}||valign="top"| F. Iversen, "Récherches sur les fonctions inverses des fonctions méromorphes", Helsinki (1914) | ||
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Latest revision as of 21:07, 31 July 2012
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) |
How to Cite This Entry:
Iversen theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iversen_theorem&oldid=12397
Iversen theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iversen_theorem&oldid=12397
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article