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We comment on Teichm{\"u}ller 's paper ''Vollst{\"a}ndige L{\"o}sung einerExtremalaufgabe der quasikonformen Abbildung'' (Complete solution of anex-tremal problem of the quasiconformal mapping),, published in 1941. In thispaper, Teichm{\"u}ller gives a proof of the existence of extremalquasiconformal mappings in the case of the pentagon (disc with fivedistinguished points on the boundary). The final version of this paper willappear as a chapter in Volume VI of the Handbook of Teichm{\"u}ller theory.

A Commentary on Teichm{\"u}ller's paper ``Vollst{\"a}ndige L{\"o}sung  einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an  extremal problem of the quasiconformal mapping)