Koebe function

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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).


[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:
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article