View source for Courant theorem

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''on conformal mapping of domains with variable boundaries''

Let $\{D_n\}$ be a sequence of nested simply-connected domains in the complex $z$-plane, $\overline{D_{n+1}}\subset D_n$, which converges to its kernel $D_{z_0}$ relative to some point $z_0$; the set $D_{z_0}$ is assumed to be bounded by a Jordan curve. Then the sequence of functions $\{w=f_n(z)\}$ which map $D_n$ conformally onto the disc $\Delta=\{w:|w|<1\}$, $f_n(z_0)=0$, $f'_n(z_0)>0$, is uniformly convergent in the closed domain $\overline{D_{z_0}}$ to the function $w=f(z)$ which maps $D_{z_0}$ conformally onto $\Delta$, moreover $f(z_0)=0$, $f'(z_0)>0$.

This theorem, due to R. Courant , is an extension of the [[Carathéodory theorem|Carathéodory theorem]].

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  R. Courant,   ''Gott. Nachr.''  (1914)  pp. 101–109</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  R. Courant,   ''Gott. Nachr.''  (1922)  pp. 69–70</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====
Cf. [[Carathéodory theorem|Carathéodory theorem]] for the definition of  "kernel of a sequence of domains" .