Koebe function
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
The function
where . This function was first studied by P. Koebe [1]. The Koebe function maps the disc onto the -plane with a slit along the ray starting at the point , its extension containing the point . 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