By C. Caratheodory

Professor Caratheodory units out the fundamental conception of conformal representations as easily as attainable. within the early chapters on Mobius' and different basic changes and on non-Euclidean geometry, he bargains with these basic matters which are precious for an realizing of the final conception mentioned within the final chapters.

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10). +ar-45r3+ .... nd v. Mises. Die Differential- und Integralgleichungen der Mechanik und Physik. Vol. 1, ch. 5. CHAPTER ill ELEMENTARY TRANSFORMATIONS 49. The ezponentlal function. The function ...... (49•1) W=e" gives rise to two important special transformations. e. +, we replace (49·1) by the two equations p = e", 4> = y. It 1- The representation is conformal throughout the interior of these regions, since the derivative of e" is never zero. J2 = '~~"), the wedge hecomes a half-plane. J1 - y2 1< 2w, may be dropped.

If n is made to increase indefinitely, the Riemann surface dealt with in §56 becomes, in the limit, the Riemann surface of §57, having a logarithmic branch-point. It is therefore to be expected that, if ~. (z} denotes the right-hand member of (56"2), and 1/J (z) that of (57"1), we .. (z} = 1/J(z). _ shall have . •... (58"1) The truth of (58"1) can in fact be deduced from a general theorem; but the equation can also be verified directly, as follows. In 1 (Jii. , I _! 6,. =0; the exponent of e in (58"2) can now be written as .......

III transforms the cut w-plane into the u-plane cut along the negative real axis, and the further transformation u = t 2 transforms this into the halfplane Jli (t) > 0. 'fo the point w = ~ correspond the points u = 1 and t = 1, so that the required transformation is obtained by writing 1 +t z=1-t" Thus finally w=~(z+D; ...... (60"1) the relation (60"1) is very remarkable in that it represents the cut wplalie not only on the ezterior but also on the interior of the unit-circle. 61. If in (60"1) we write z=re;', where (for example) r> 1, we obtain w=~{(r+~)cos8+i(r-Dsin0}; ......