By Carlos A. Berenstein, Roger Gay

Textbooks, even first-class ones, are a mirrored image in their occasions. shape and content material of books depend upon what the scholars comprehend already, what they're anticipated to benefit, how the subject material is appeared when it comes to different divisions of arithmetic, or even how stylish the subject material is. it's hence no longer impressive that we not use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our classes. The final twenty years have visible an important switch within the options utilized in the idea of capabilities of 1 complicated variable. the real position performed via the inhomogeneous Cauchy-Riemann equation within the present study has resulted in the reunification, not less than of their spirit, of advanced research in a single and in different variables. we are saying reunification when you consider that we predict that Weierstrass, Poincare, and others (in distinction to a lot of our scholars) didn't give some thought to them to be totally separate topics. certainly, not just complicated research in numerous variables, but additionally quantity idea, harmonic research, and different branches of arithmetic, either natural and utilized, have required a reconsidera­tion of analytic continuation, usual differential equations within the complicated area, asymptotic research, new release of holomorphic services, and lots of different matters from the vintage idea of services of 1 advanced variable. This ongoing reconsideration led us to imagine textbook incorporating a few of these new views and strategies needed to be written.

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R is such that Up n 0 = g E Up: pp(O < O} and Up n 00 = g E Up: pp(O = O}. Furthermore, I/Ip:= o-pl(Up n (0): Up n 00 ...... t;, n {(x, Y) E [R2 : x = O} is a homeomorphism onto J -1, 1[ such that I/Ip-l : J -1,1 [ ...... 0 is of class C k • Let ( = (~, '1) and Y = I/I q«()' Since Pp 0 I/Ip-l == 0 in ] -I, I[ one sees that the vector grad pp(O = (a p! (0, app (0) is orthogonal to (1/1;'1 ),(y), the tangent vector to or:; 0'1 23 §4. Regular Boundaries the curve Up 11 an parameterized by t/Jp-l.

Remark. More generally, let (Y;)1 ,;;i';;n be a family of paths such that Yi(I) = Yi+1(0)(I $; i $; n - 1) and let 0 = to < t1 < ... < tn = 1 be a partition of [0, 1]. -l ti ti- 1 is homotopic to the path Y1(Y2('" Yn) ... ). Moreover, if IXi is homotopic to Yi, the corresponding path IX is homotopic to y. Therefore, we can talk about the path Yl ... Yn "up to homotopy," without worrying about placing parentheses. Obviously the same is true for the composition of the homotopy classes [Yl]'" [Yn].

PROOF. Let us first prove the uniqueness of the lifting c of y. Let x E [0, 1] and c l , C 2 two liftings ofy defined over [0, x] such that c1(0) = c2 (0). Then we have 1 = ec ,(t)-C2(t) for ~ t ~ x . (O) = 0, we have c, = c? xl. Let us now prove the existence of a lifting c of y. Let I = {x E [0,1]: 3 cx : [0, x] -+ C, continuous, cx(O) = zo, and expc x = yl[O, x]}. The set I is not empty because E I. Let x E I and ~ x' ~ x, then Cx' = cXI[O,x /] has all the necessary properties, hence x' E I.

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