By Florian Cajori

ISBN-10: 0486621847

ISBN-13: 9780486621845

ISBN-10: 1418165557

ISBN-13: 9781418165550

Initially released in 1904. This quantity from the Cornell college Library's print collections was once scanned on an APT BookScan and switched over to JPG 2000 layout through Kirtas applied sciences. All titles scanned disguise to hide and pages could comprise marks notations and different marginalia found in the unique quantity.

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1 'Definition ([PI], cf. [L/V], pp. 168-170). Let G g C be a domain, and K a real constant ^ 1 . e. i n G , k = { K - l)l(K + 1). 13 the 1-quasiconformal mappings are just the conformal mappings. e. in G, cf. 29. 2 Theorem. e. in G, belongs to fVhoc(G \ {oo,/-^oo)}) and, therefore, is absolutely continuous {as a homeomorphism o f G \ {oo,/“^(oo)}, cf. 2 above). 20 we only have to prove th at/b e lo n g s to H lo c (G \ {oo,/“^(oo)}). Let G' be any domain compactly contained in G \ {oo,/“‘(oo)}.

168-170). Let G g C be a domain, and K a real constant ^ 1 . e. i n G , k = { K - l)l(K + 1). 13 the 1-quasiconformal mappings are just the conformal mappings. e. in G, cf. 29. 2 Theorem. e. in G, belongs to fVhoc(G \ {oo,/-^oo)}) and, therefore, is absolutely continuous {as a homeomorphism o f G \ {oo,/“^(oo)}, cf. 2 above). 20 we only have to prove th at/b e lo n g s to H lo c (G \ {oo,/“^(oo)}). Let G' be any domain compactly contained in G \ {oo,/“‘(oo)}. Then r(e) := \f{er\G ')\ is a finite RBM, cf.

E. in G, (II) f e Wpjo/G) for each p > 2 for which K - 1 C(p) < 1, K+ 1 (III) / is Hôlder continuous in every domain G' ê G. 2. SCHLICHT BELTRAMI SOLUTIONS 31 Indeed let v(z) = /¡If, i ff , #= 0, z e G, and v(z) = 0 else. Then/(z) = F o f \z ) , F analytic i n / '’(G). This proves (I) and (II). To prove (III) we put v(z) = /¡¡f, in G", G' € G" € G, and r(z) = 0 outside G". T hen/(z) = H o p - \z ) , H an alytic in /''■'’(G"). 10 the assertion (III) follows. 30 The inverse mapping o f a K-quasiconformal mapping is a K-quasiconformal mapping, too.

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