Conformal Representation by C. Caratheodory

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}; ......

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