View source for Koebe function

The function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556501.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556502.png" />. This function was first studied by P. Koebe [[#References|[1]]]. The Koebe function maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556503.png" /> onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556504.png" />-plane with a slit along the ray starting at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556505.png" />, its extension containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055650/k0556506.png" />. 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====
<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>