By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

ISBN-10: 3834815764

ISBN-13: 9783834815767

This rigorously written textbook is an advent to the gorgeous options and result of advanced research. it's meant for foreign bachelor and grasp programmes in Germany and all through Europe; within the Anglo-American approach of college schooling the content material corresponds to a starting graduate path. The e-book provides the elemental effects and strategies of complicated research and applies them to a examine of uncomplicated and non-elementary services elliptic features, Gamma- and Zeta functionality together with an evidence of the leading quantity theorem ' and ' a brand new characteristic during this context! ' to showing uncomplicated evidence within the concept of numerous complicated variables. a part of the publication is a translation of the authors' German textual content 'Einfuhrung in die komplexe Analysis'; a few fabric used to be further from the by means of now nearly 'classical' textual content 'Funktionentheorie' written through the authors, and some paragraphs have been newly written for precise use in a master's programme. content material research within the complicated aircraft - the elemental theorems of advanced research - capabilities at the aircraft and at the sphere - imperative formulation, residues and purposes - Non-elementary features - Meromorphic capabilities of numerous variables - Holomorphic maps: Geometric facets Readership complicated undergraduates bachelor scholars and starting graduate scholars master's programme teachers in arithmetic in regards to the authors Professor Dr. Ingo Lieb, division of arithmetic, collage of Bonn Professor Dr. Wolfgang Fischer, division of arithmetic, collage of Bremen

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If γ is a path of integration in the open set U ⊂ C and h : U → V is a holomorphic function, then h ◦ γ is a path of integration in V . It follows that if f is continuous on V , then b a h◦γ (f ◦ h)h (z) dz. f h γ(t) h γ(t) γ (t) dt = f (w) dw = (11) γ Note the extra factor h that appears in the integrand! Paths of integration may, in a natural way, be combined, subdivided, and reversed: a) Let γ1 and γ2 be two paths of integration such that the end point of γ1 is the initial point of γ2 . We may – reparametrizing if necessary – assume that γ1 and γ2 are deﬁned on the intervals [a, b] and [b, c].

Tn ) = (γ1 (t1 ), . . , γn (tn )) deﬁnes a mapping of the n-dimensional cube Qn = {(t1 , . . , tn ) : 0 tν 1} ⊂ Rn ν n. Then (7) 6. Several complex variables 35 into Cn . We call Γ a parametrized surface of integration in Cn ; its image Γ(Qn ) is the trace of Γ. For a continuous function f on the trace of Γ, we deﬁne ··· f (z1 , . . , zn ) dz1 . . dzn = γn Γ f (z1 , . . , zn ) dz1 dz2 . . dzn . γ2 (8) γ1 Thus one ﬁrst ﬁxes the variables z2 , . . , zn and integrates the function z1 → f (z1 , z2 , .

Then f is holomorphic on G. Proof: Let D be an arbitrary disk in G. Given our assumptions, Prop. 3 implies that f has a primitive F on D. By Thm. 4, f = F is holomorphic on D. Remark: Let f : G → C be continuous on the domain G and holomorphic on G\{z0 } for some z0 ∈ G. 2 that ∂Δ f (z) dz = 0 for every closed triangle Δ in G, so that f is holomorphic on all of G, by Morera’s theorem. The condition “continuous, and holomorphic with the possible exception of one point” thus only seems to be more general than the condition “holomorphic”.

### A Course in Complex Analysis: From Basic Results to Advanced Topics by Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

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