By L. Bieberbach

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LINEAR FUNCTIONS inverse with respect to K lies on a circle perpendicular to K, the points of the image pair must be mutually inverse with respect to the image of K, which is what we wished to prove. 4. , of the problem of determining all linear mappings of a circular disc onto itself, acquires an even greater importance through the fact, to be proved presently, that there are no other one-to-one conformal mappings of a circular disc onto itself. To prove this, it will suffice to prove that all one-to-one conformal mappings of the circular disc I z I < 1 that leave its center z ~ 0 fixed, are linear.

RATIONAL FUNCTIONS decomposition of the Riemann surface into n sheets each of which covers the w-plane. Such different systems would correspond to decompositions of the z-plane into sectors different from the ones used above. , w = zn) at w = 0 and w = 00. Let us now ask, what about the Preservation-of-Neighborhoods Theorem in connection with our function? It obviously applies without any modification to the neighborhood of any point other than zero and infinity. Peculiarities are encountered, however, when we deal with the mapping of neighborhoods of z = 0 and z = 00, or with the mapping of the whole z-plane.

A fundamental region for this group, shaded in Fig. 8, is the part outside the unit circle of the strip between the two lines x === -1/2 and I I I ' I I I \ x= + 1/2 (z=x + iy). For the proof, we refer the reader to more detailed exposiFIG. 8 tions (such as Vol. II of the author's Lehrbuch der Funktio1tentheorie, Chelsea Pub!. , New York 1945). Here we merely add that all the mappings of the group can be generated by composition from two of them, namely from the parabolic mapping z' = z + 1 and the elliptic mapping w = -l/z.