By C. Caratheodory

Professor Caratheodory units out the fundamental thought of conformal representations as easily as attainable. within the early chapters on Mobius' and different basic adjustments and on non-Euclidean geometry, he bargains with these user-friendly topics which are valuable for an knowing of the overall concept mentioned within the final chapters.

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**Extra info for Conformal Representation, 2nd Edition **

**Example text**

56"2) = ...... e- z)" w=h' w = k ' 1 16 • (564) • ••••• (56"4) 57. The Thecase case in in which which the the Riemanu Riemann surface has aa logarithmic logarithmic branchbranchhe treated in point at atto0 w 0 == kii may be in aa similar similar manner. manner. We Wefirst, first, by by means means transform this Riemaun Riemann surface into the surface already conof (56"1), transform virtue of of the thesame same paragraph, paragraph, the the transforinatransformain §§ 50; then, in virtue sidered in et transforms this surface = e' tion u = surface into the the half-plane half-plane lt (t) <<0, 0, making making A.

Hence, Hence, by by the the foreforegoing, each each of ofthese these crescent-shaped crescent-shaped areas areas can be be represented represented conformally circle. In In the the same same way way the exterior exterior of of aa circular circular on the interior of of a circle. Pig. 18 Fig. 18 crescent or the the exterior exterior of of two two circles circles having external contact can also be represented on the interior interior of of aa circle. circle. 55. 55. Representation Representation of ofRlemann Riemann surfaces.

By transformation u == vv44 we we now now obtain different By means means of of the the transformation obtain four different domains in in the the v-plane v-planecorresponding correspondingtotoT,T1; let TT2 denote that one ot domains ; let denote that 2 v > 0 corresponds corresponds to straight line line them them in which whichthe the straight straight line line v> to the straight (v) >0, and is u>O*. { (v)>O, u> 0*. ••• (64'1) (64-1) v+1 T' which into a domain T' which lies inside the circle circle Ij w T 2 were were to wI<< 1.