By Vasiliy Sergeyevich Vladimirov, Mathematics

ISBN-10: 0486458121

ISBN-13: 9780486458120

This systematic exposition outlines the basics of the idea of unmarried sheeted domain names of holomorphy. It extra illustrates purposes to quantum box idea, the idea of capabilities, and differential equations with consistent coefficients. scholars of quantum box conception will locate this article of specific price. 1966 variation.

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**Example text**

Then, if we differentiate the equation /(z)/(z) = M2 and apply the Cauchy-Riemann conditions oflo"Zj, for j = 1. 2..... , we obtain = - df 0/ - of Of - df OZj iJZj oZJ OZj oZJ /-+/-=/-+/-=-=/-=0. /=1. 2..... n. Since 171 = iJ/liJzj=o. M > 0, this last relationship implies the equations J= 1. 2..... n. These equations, together with the Cauchy-Riemann conditions yield / (z) = const in S (zO. r). By virtue of the holomorphic continuation theorem / (z) const in O. But this is ruled out by the hypotheses of the theorem.

Then, these functions are holomorphic continuation each of the other; that is, there exists a unique function f(z) that is holomorphic in GI U G2 and that coincides with fl (z) in Gl and with f2 (z) in G2. Proof: It will be sufficient to show that II (z'l==/2 (z) for z E0 1 O2• In the largest polycircle S(ZO. ro/), contained in 01 n02. 4) I,,(z)= ~ a~")(z-zO)". I D·I" (zO). k == 1. 2. (60) 1_1>0 If we take the derivatives in the real neighborhood S(XO. r), y=YJ. where. by hypothesis. II(x+lyO)=/2(X+'YJ).

00 cpE D (0). Since the functions /. (z) satisfy the Cauchy-Riemann conditions and the differential operators are continuous from D'(O) into D' (O) (cf. 2). {z)-+O"/{z) uniformly in O. 'Suppose that S(ZO, rl)<60. Then, if we apply formula (41) to the functions /. (z) and remember (see Fig. 3) that S(zo, -i/)US(z" f/)cS{ZO, rI), S -i I) X S(zOo -i I) Fig. 3 we obtain for all(z, z')E (zOo D"I ( ) J .. I • z = (2lti)n h. ,~r) Suppose that the function f cp (z') dx' dy' = cp{z'):=cp(x', y') belongs to O[S(O, r/41)1 and 1.

### Methods of the Theory of Functions of Many Complex Variables by Vasiliy Sergeyevich Vladimirov, Mathematics

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