Koebe function

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)