Namespaces
Variants
Actions

Difference between revisions of "Iversen theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX, Refs, MSC (please check MSC))
 
Line 1: Line 1:
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530501.png" /> is an isolated essential singularity of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530502.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530503.png" />, then every [[Exceptional value|exceptional value]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530504.png" /> in the sense of E. Picard is an [[Asymptotic value|asymptotic value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530505.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530506.png" />. For example, the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530508.png" /> are exceptional and asymptotic values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i0530509.png" /> at the essential singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i05305010.png" />. This result of F. Iversen [[#References|[1]]] supplements the big [[Picard theorem|Picard theorem]] on the behaviour of an analytic function in a neighbourhood of an essential singularity.
+
{{MSC|32H25}}
 +
{{TEX|done}}
  
====References====
+
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.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Iversen,   "Récherches sur les fonctions inverses des fonctions méromorphes" , Helsinki  (1914)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR></table>
 
  
 +
Iversen's theorem has been extended to subharmonic functions on $\R^n$, notably by W.K. Hayman, see {{Cite|HaKe}}, {{Cite|Ha}}.
  
 
+
====References====  
====Comments====
+
{|
Iversen's theorem has been extended to subharmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053050/i05305011.png" />, notably by W.K. Hayman, see [[#References|[a1]]], [[#References|[a2]]].
+
|-
 
+
|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
====References====
+
|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"W.K. Hayman,   P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"W.K. Hayman,   "Subharmonic functions" , '''2''' , Acad. Press (1989)</TD></TR></table>
+
|valign="top"|{{Ref|Ha}}||valign="top"| W.K. Hayman, "Subharmonic functions", '''2''', Acad. Press (1989)
 +
|-
 +
|valign="top"|{{Ref|HaKe}}||valign="top"| W.K. Hayman, P.B. Kennedy, "Subharmonic functions", '''1''', Acad. Press (1976)
 +
|-
 +
|valign="top"|{{Ref|Iv}}||valign="top"| F. Iversen, "Récherches sur les fonctions inverses des fonctions méromorphes", Helsinki (1914)
 +
|-
 +
|}

Latest revision as of 23:07, 31 July 2012

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