Namespaces
Variants
Actions

Difference between revisions of "Koebe function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
m (typo)
 
(One intermediate revision by one other user not shown)
Line 4: Line 4:
 
$$w=f(z)=f_\theta(z)=\frac{z}{(1-e^{i\theta}z)^2}=z+\sum_{n=2}^\infty ne^{i(n-1)\theta}z^n,$$
 
$$w=f(z)=f_\theta(z)=\frac{z}{(1-e^{i\theta}z)^2}=z+\sum_{n=2}^\infty ne^{i(n-1)\theta}z^n,$$
  
where $\theta\in[0,2\pi)$. This function was first studied by P. Koebe [[#References|[1]]]. The Koebe function maps the disc $|z|<1$ onto the $w$-plane with a slit along the ray starting at the point $-e^{-i\theta}/4$, its extension containing the point $w=0$. The Koebe function is an extremal function in a number of problems in the theory of univalent functions (cf. [[Bieberbach conjecture|Bieberbach conjecture]]; [[Univalent function|Univalent function]]).
+
where $\theta\in[0,2\pi)$. This function was first studied by P. Koebe [[#References|[1]]]. The Koebe function maps the disc $|z|<1$ onto the $w$-plane with a slit along the ray starting at the point $-e^{-i\theta}/4$, its extension containing the point $w=0$. The Koebe function is an extremal function in a number of problems in the theory of univalent functions (cf. [[Bieberbach conjecture]]; [[Univalent function]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Koebe,   "Ueber die Uniformisierung beliebiger analytischen Kurven"  ''Math. Ann.'' , '''69'''  (1910)  pp. 1–81</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.K. Hayman,   "Coefficient problems for univalent functions and related function classes"  ''J. London Math. Soc.'' , '''40''' :  3  (1965)  pp. 385–406</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  P. Koebe, "Über die Uniformisierung beliebiger analytischen Kurven"  ''Math. Ann.'' , '''69'''  (1910)  pp. 1–81</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  W.K. Hayman, "Coefficient problems for univalent functions and related function classes"  ''J. London Math. Soc.'' , '''40''' :  3  (1965)  pp. 385–406</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR>
 +
</table>
 +
 
 +
[[Category:Functions of a complex variable]]

Latest revision as of 19:36, 5 April 2023

The function

$$w=f(z)=f_\theta(z)=\frac{z}{(1-e^{i\theta}z)^2}=z+\sum_{n=2}^\infty ne^{i(n-1)\theta}z^n,$$

where $\theta\in[0,2\pi)$. This function was first studied by P. Koebe [1]. The Koebe function maps the disc $|z|<1$ onto the $w$-plane with a slit along the ray starting at the point $-e^{-i\theta}/4$, its extension containing the point $w=0$. The Koebe function is an extremal function in a number of problems in the theory of univalent functions (cf. Bieberbach conjecture; Univalent function).

References

[1] P. Koebe, "Über die Uniformisierung beliebiger analytischen Kurven" Math. Ann. , 69 (1910) pp. 1–81
[2] W.K. Hayman, "Coefficient problems for univalent functions and related function classes" J. London Math. Soc. , 40 : 3 (1965) pp. 385–406
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Koebe function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Koebe_function&oldid=33344
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article