Difference between revisions of "Koebe function"
From Encyclopedia of Mathematics
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The function | 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 | + | 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]]). |
====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, "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> | ||
Revision as of 09:57, 20 September 2014
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, "Ueber 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=19034
Koebe function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Koebe_function&oldid=19034
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article